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mm LIBRARY OF CONGRESS. 



wm Lit 

4g> 9-1 ( 



A TEXTBOOK 



ON 



HYDRAULIC ENGINEERING 

International Correspondence Schools 

SCRANTON, PA. 



ANSWERS TO QUESTIONS 
TABLES AND FORMULAS 



SCRANTON 
INTERNATIONAL TEXTBOOK COMPANY 



A-4 



; 



THt UOHARY OF 

CONGRESS. 

One Co<=' Reoeiveo 

JUL t5 »^04 

Oo*>vRIGHT BWTRV 

**e. No. 

3 IT if 

COPY A. 



/fRA^O. 



Copyright, 1897, 1898, by The Colliery Engineer Company. 



Arithmetic, Key : Copyright, 1S93, 1894, 1896, 1897, 1898, by THE COLLIERY Engi- 
neer Company. 

Algebra, Key: Copyright, ; l'S94, 1S9G, 1897, 1S98, by THE COLLIERY ENGINEER 
Company. T 

Logarithms, Key : Copyright/1897, by The Colliery Engineer Company. 

Geometry and Trigonometry^ Key : Copyright, 1893, 1894, 1895, 1898, by The Col- 
liery Engineer Compaq. 

Elementary Mechanics, Key *• Copyright, 1893, 1894, 1835, 1837, by The Colliery 
Engineer Company. * 

Hydraulics, Key : Copyright^ 1893, 1894, 1895, 1897, by The Colliery Engineer 
Company. * *-, 

Pneumatics, Key: Copyright,- 1893£ £§94, 1895, 1897, by THE COLLIERY ENGINEER 
Company. -.** ' i 

Strength of Materials, Key: Copyright, 1894, by The Colliery Engineer Com- 
pany. •* 

Surveying, Key : Copyright,, 1895, Jby. The Colliery Engineer Company. 

Surveying and Mapping. Key: Copyright, 1898,' by THE COLLIERY ENGINEER 
Company. '■ ... ._-v 

Steam and Steam Engines, Key: Copyright,'';i894 T 1895, 1898, by THE COLLIERY 
Engineer Company. 

Steam Boilers, Key: Copyright^, 1895, 1898, by The Colliery Engineer Com- 
pany. 

Water- Wheels, Key. Copyright 1898, by The Colliery Engineer Company. 

Hydraulic Machinery, Key: .Copyright. 1S97, by THE COLLIERY ENGINEER COM- 
PANY. 

Water Supply and Distribution, Key: 'Copyright, 1898. by The Cclliery Engi- 
neer Company. 

Irrigation, Key: Copyright, 1898, by THE : -QOLLIERY ENGINEER Company. 

Tables and Formulas: Copyright. 1897, 1898, by The Colliery ENGINEER 
Company. -■ • 



All rights reserved. 



BURR PRINTING' HOUSE,- 
FRANKFORT AND JACOB ST 



NEW YORK. 



REETS, p A 



, 






TMP96-024524 



A KEY 

TO ALL THE 

QUESTIONS AND EXAMPLES 

CONTAINED IN THE 

EXAMINATION QUESTIONS 

Included in the Preceding Volumes. 



It will be noticed that the Key is divided into sections 
that correspond to the sections containing the questions and 
examples at the end of the preceding volumes. The answers 
and solutions are so numbered as to be similar to the num- 
bers before the questions to which they refer. When the 
answer to a question involves a repetition of statements 
given in the text, the reader has been referred to a num- 
bered article, the reading of which will enable him to answer 
the question himself. 

To be of the greatest benefit, the Keys should be used 
sparingly. They should be used much in the same manner 
as a pupil would go to a teacher for instruction with regard 
to answering some example he was unable to solve. If used 
in this manner, the Keys will be of great help and assist- 
ance to the student, and will be a source of encouragement 
to him in studying the various papers composing the Course. 



CONTENTS. 



Arithmetic, ... - Answers 

Algebra, ----- Answers 

Logarithms, - Answers 

Geometry and Trigonometry, - Answers 

Elementary Mechanics, - - Answers 

Hydraulics, - Answers 

Pneumatics, ... - Answers 

Strength of Materials, - - Answers 

Surveying, - Answers 

Surveying and Mapping, - Answers 

Steam and Steam Engines, - Answers 

Steam Boilers, - Answers 

Water-Wheels, - - - Answers 

Hydraulic Machinery, - - Answers 

Water Supply and Distribution, Answers 

Irrigation, - Answers 



PAGE. 

to Questions 1 

to Questions 95 

to Questions 137 

to Questions 145 

to Questions 165 

to Questions 187 

to Questions 207 

to Questions 215 

to Questions 239 

to Questions 263 

to Questions 275 

to Questions 297 

to Questions 307 

to Questions 317 

to Questions 327 

to Questions 347 



ARITHMETIC 

(SECTIONS 1-3.) 
(QUESTIONS 1-75.) 



(1) See Art. 1. 

(2) See Art. 3. 

(3) See Arts. 5 and 6. 

(4) See Arts. lO and 11. 

(5) 980 = Nine hundred eighty. 
605 = Six hundred five. 

28,284 = Twenty-eight thousand, two hundred eighty-four, 
9,006,042 = Nine million, six thousand and forty-two. 
850,317,002= Eight hundred fifty million, three hundred 
seventeen thousand and two. 

700,004 = Seven hundred thousand and four. 

(6) Seven thousand six hundred = 7,600. 
Eighty-one thousand four hundred two = 81,402. 
Five million, four thousand and seven = 5,004,007. 

One hundred and eight million, ten thousand and one =. 
108,010,001. 

Eighteen million and six = 18,000,006. 
Thirty thousand and ten = 30,010. 

(7) In adding whole numbers, place the numbers to be 
added directly under each other so that 3 2 9 

the extreme right-hand figures will stand ~ q ^ 

in the same column, regardless of the q G 5 4 3 

position of those at the left. Add the first l > 7 4 

column of figures at the extreme right, g -^ 

which equals 19 units, or 1 ten and 9 « 

units. We place 9 units under the units 

column, and reserve 1 ten for the column M ' ° l 

For notice of the copyright, see page immediately following the title page 



2 ARITHMETIC. 

of tens. 1 + 8 + 7 + 9 = 25 tens, or 2 hundreds and 5 
tens. Place 5 tens under the tens column, and reserve 
2 hundreds for the hundreds column. 2 + 4+5 + 2 = 13 
hundreds, or 1 thousand and 3 hundreds. Place 3 hundreds 
under the hundreds column, and reserve the 1 thousand 
for the thousands column. 1 + 2 + 5 + 3 = 11 thousands, 
or 1 ten-thousand and 1 thousand. Place the 1 thousand in 
the column of thousands, and reserve the 1 ten-thousand 
for the column of ten-thousands. 1 + 6 = 7 ten-thousands. 
Place this seven ten-thousands in the ten-thousands column. 
There is but one figure 8 in the hundreds of thousands place 
in the numbers to be added, so it is placed in the hundreds 
of thousands column of the sum. 

A simpler (though less scientific) explanation of the same 
problem is the following : 7+1 + 4+3 + 4+0 = 19; write 
the nine and reserve the 1. 1 + 8+7 + + + 9=25; 
write the 5 and reserve the 2. 2 + + 4 + 5 + 2=13: 
write the 3 and reserve the 1. 1 + 2 + 5 + 3 = 11; write 
the 1 and reserve 1. 1+6 = 7; write the 7. Bring down 
the 8 to its place in the sum. 



(8) 



709 


8304725 


391 


100302 


300 


909 


8407336 



Ans. 



(9) (a) In subtracting whole numbers, place the sub- 
trahend or smaller number under the minuend or larger 
number, so that the right-hand figures stand directly under 
each other. Begin at the right to subtract. We can not 
subtract 8 units from 2 units, so we take 1 ten from the 
6 tens and add it to the 2 units. As 1 ten = 10 units, we 
have 10 units + 2 units = 12 units. Then, 8 units from 
12 units leaves 4 units. We took 1 ten from 6 tens, so 



ARITHMETIC. 

only 5 tens remain. 3 tens from 5 tens 59952 
leaves 2 tens. In the hundreds column we 3338 
have 3 hundreds from 9 hundreds leaves 



6 hundreds. We can not subtract 3 thou- 4 7 G 2 4 Ans * 
sands from thousands, so we take 1 ten-thousand from 
5 ten-thousands and add it to the thousands. 1 ten- 
thousand = 10 thousands, and 10 thousands -f- thousands 
= 10 thousands. Subtracting, we have 3 thousands from 
10 thousands leaves 7 thousands. We took 1 ten-thousand 
from 5 ten-thousands and have 4 ten-thousands remaining. 
Since there are no ten-thousands in the subtrahend, the 
4 in the ten-thousands column in the minuend is brought 
down into the same column in the remainder, because from 
4 leaves 4. 

(b) 1533 9 
10001 



5338 Ans. 

(IO) (a) 70968 (b) 100000 

32975 98735 



3 7993 Ans. 1265 Ans. 

(11) We have given the minuend or greater number 
(1,004) and the difference or remainder (49). Placing these 

1004 

in the usual form of subtraction we have in which 

49 

the dash ( ) represents the number sought. This number 

is evidently less than 1,004 by the difference 49, hence, 
1,004 — 49 = 955, the smaller number. For the sum of the 

10 4 larger 
two numbers we then have 9 5 5 smaller 

19 5 9 sum. Ans. 

Or, this problem may be solved as follows: If the greater 

of two numbers is 1,004, and the difference between them is 

49, then it is evident that the smaller number must be 

equal to the difference between the greater number (1,004) 



4 ARITHMETIC. 

and the difference (49); or, 1,004 — 49 = 955, the smaller 

number. Since the greater number equals 1,004 and the 

smaller number equals 955, their sum equals 1,004 + 955 
= 1,959 sum. Ans. 

(12) The numbers connected by the plus (+) sign must 
first be added. Performing these operations we have 
5962 3874 

8471 20 3 9 

9Q 23 5 913 sunta 

2 3 4 5 6 sum. 
Subtracting the smaller number (5,913) from the greater 
(23,456) we have 

23456 
5913 



17 5 4 3 difference. Ans. 

(13) $44675 = amount willed to his son. 

2 6 3 8 = amount willed to his daughter. 
$71055 = amount willed to his two children. 
$12 5 000 = amount willed to his wife and two 
children. 
710 5 5 = amount willed to his two children. 
$53945 = amount willed to his wife. Ans. 

(14) In the multiplication of whole numbers, place the 
multiplier under the multiplicand, and multiply each term 
of the multiplicand by each term of the multiplier, writing 
the right-hand figure of each product obtained under the 
term of the multiplier which produces it. 

(a) 7x7 units = 49 units, or 4 tens and 9 

52 6 3 87 units. We write the 9 units and reserve 

7 the 4 tens. 7 times 8 tens = 56 tens; 

3 6 8 4 7 9 Ans. 56 tens -J- 4 tens reserved = 00 tens or 
6 hundreds and tens. Write the 
tens and reserve the 6 hundreds. 7 X 3 hundreds = 21 hun- 
dreds; 21 + 6 hundreds reserved = 27 hundreds, or 2 thou- 
sands and 7 hundreds. Write the 7 hundreds and reserve 



ARITHMETIC. 5 

the 2 thousands. 7xG thousands = 42 thousands; 42 
+ 2 thousands reserved = 44 thousands or 4 ten-thousands 
and 4 thousands. Write the 4 thousands and reserve the 
4 ten-thousands. 7x2 ten-thousands = 14 ten-thousands; 
14 + 4 ten-thousands reserved — 18 ten-thousands, or 
1 hundred-thousand and 8 ten-thousands. Write the 8 ten- 
thousands and reserve the 1 hundred-thousand. 7x5 hun- 
dred-thousands = 35 hundred-thousands; 35 -j- 1 hundred- 
thousand reserved = 3G hundred-thousands. Since there 
are no more figures in the multiplicand to be multiplied, 
we write the 3G hundred-thousands in the product. This 
completes the multiplication. 

A simpler (though less scientific) explanation of the same 
problem is the following: 

7 times 7 = 49 ; write the 9 and reserve the 4. 7 times 
8 = 5G ; 56 + 4 reserved = GO ; write the and reserve the 6. 
7 times 3 = 21 ; 21 + 6 reserved = 27; write the 7 and re- 
serve the 2. 7 X 6 = 42; 42 + 2 reserved =. 44; write the 
4 and reserve 4. 7 X 2 = 14; 14 + 4 reserved = 18; write 
the 8 and reserve the 1. 7 X 5 = 35; 35 + 1 reserved = 36; 
write the 36. 

In this case the multiplier is 17 
units, or 1 ten and 7 units, so that (P) 7 2 9 8 

the product is obtained by adding 1 7 

two partial products, namely, 7X 4902086 

700,298 and 10 X 700,298. The 700298 

actual operation is performed as 11905066 Ans. 

follows: 

7 times 8 = 56; write the 6 and reserve the 5. 7 times 9 = 
63; 63 + 5 reserved = 68 ; write the 8 and reserve the 6. 
7 times 2 = 14; 14+6 reserved = 20; write the and re- 
serve the 2. 7 times = 0; + 2 reserved = 2 ; write the 2. 
7 times = ; + reserved = ; write the 0. 7 times 7 = 
49 ; 49 + reserved = 49 ; write the 49. 

To multiply by the 1 ten we say 1 times 700298 = 700298, 
and write 700298 under the first partial product, as shown, 
with the right-hand figure 8 under the multiplier 1. Add the 
two partial products ; their sum equals the entire product. 






ARITHMETIC. 

(c) 217 Multiply any two of the numbers together 

103 and multiply their product by the third 

6 5 1 number, 
2170 



22351 
67 

156457 
134106 
14 9 7 517 Ans. 

(15) If your watch ticks every second, then to find how 
many times it ticks in one week it is necessary to find the 
number of seconds in 1 week. 

6 seconds = 1 minute. 
6 minutes = 1 hour. ' 
3 6 seconds = 1 hour. 
2 4 hours = 1 day. 



14400 
7200 



8 6 4 seconds = 1 day. 
7 days = 1 week. 



6 4 8 seconds in 1 week or the number of times that 
Ans. your watch ticks in 1 week. 

(16) If a monthly publication contains 24 pages, a yearly 

2 4 volume will contain 12x24 or 288 pages, since 

12 there are 12 months in one year; and eight 

2g g yearly volumes will contain 8x288, or 2,304 

o pages. 



2 3 4 Ans. 

(17) If an engine and boiler are worth $3,246, and the 
building is worth 3 times as much, plus $1,200, then the 
building is worth 

$3246 

3 

9738 
plus 12 

$10 9 3 8 = value of building. 



ARITHMETIC. 7 

If the tools are worth twice as much as the building, plus 
$1,875, then the tools are worth 

$10938 

2 



21876 
plus 18 7 5 



$23751 = value of tools. 
Value of building = $10938 
Value of tools = 2 3 7 51 



$34689 = value of the building 
and tools, (a) Ans. 
Value of engine and 

boiler = $ 3 2 46 
Value of building 

and tools = 3 4 6 8 9 



$ 3 7 9 3 5 = value of the whole 
plant, (b) Ans. 

(18) (a) (72 X 48 X 28 X 5) -- (96 X 15 X 7 X 6). 
Placing the numerator over the denominator the problem 
becomes 

72 X 48 X 28 X 5 , 
96 X 15 X 7 X 6 * 

The 5 in the dividend and 15 in the divisor are both divis- 
ible by 5, since 5 divided by 5 equals 1, and 15 divided by 
5 equals 3. Cross off the 5 and write the 1 over it ; also cross 
off the 15 and write the 3 under it. Thus, 

1 

72 x 48 X 28 X ft 
96 x # X 7 X 6 
3 

The 5 and 15 are not to be considered any longer, and, in 
fact, may be erased entirely and the 1 and 3 placed in their 
stead, and treated as if the 5 and 15 never existed. Thus, 

72 X 48 X 28 X 1 _ 
96 X 3 X 7 X 6 



8 ARITHMETIC. 

72 in the dividend and 96 in the divisor are divisible by 12, 
since 72 divided by 12 equals 6, and 96 divided by 12 equals 
8. Cross off the 72 and write the 6 over it ; also, cross off 
the 96 and write the 8 under it. Thus, 

6 

t% X 48 X 28 X 1 _ 

?0 X 3 x 7 x 6 ~" 

8 
The 72 and 96 are not to be considered any longer, and, 
in fact, may be erased entirely and the 6 and 8 placed in 
their stead, and treated as if the 72 and 96 never existed. 
Thus, 

6 X 48 X 28 X 1 = 
8X3X7X6 
Again, 28 in the dividend and 7 in the divisor are divisible 
by 7, since 28 divided by 7 equals 4, and 7 divided by 7 
equals 1. Cross off the 28 and write the 4 over it; also, cross 
off the 7 and write the 1 under it. Thus, 

4 

6 x 48 x ffi X 1 _ 
8 X 3 X J X 6 
1 

The 28 and 7 are not to be considered any longer, and, in 
fact, may be erased entirely and the 4 and 1 placed in their 
stead, and treated as if the 28 and 7 never existed. Thus, 

6 X 48 X 4 X 1 
8X3X1X6 "~ 
Again, 48 in the dividend and 6 in the divisor are divisible 
by 6, since 48 divided by 6 equals 8, and 6 divided by 6 equals 
1. Cross off 'the 48 and write the 8 over it; also, cross off 
the 6 and write the 1 under it. Thus, 

8 

6x48x4x1 



8x 3X1X0 
1 
The 48 and 6 are not to be considered any longer, and, in 
fact, may be erased entirely and the 8 and 1 placed in their 
stead, and treated as if the 48 and 6 never existed. Thus, 



ARITHMETIC. 9 

6X8X4X1 ^ 

8X3X1X1" 
Again, 6 in the dividend and 3 in the divisor are divisible 
by 3, since G divided by 3 equals 2, and 3 divided by 3 equals 
1. Cross off the 6 and write the 2 over it; also, cross off the 
3- and write the 1 under it. Thus, 

2 

0X 8x4xl = 

8x?XlXl 
1 

The G and 3 are not to be considered any longer, and, in 
fact, may be erased entirely and the 2 and 1 placed in their 
stead, and treated as if the G and 3 never existed. Thus, 

2X8X4X1 ^ 
8X1X1X1 

Canceling the 8 in the dividend and the 8 in the divisor, 
the result is 

1 
2x$x4xl _ 2xlx4xl 

£ x l x l x l lxlxlxl' 
1 

Since there are no two remaining numbers (one in the 
dividend and one in the divisor) divisible by any number ex- 
cept 1, without a remainder, it is impossible to cancel further. 

Multiply all the uncanceled numbers in the dividend 
together, and divide their product by the product of all 
the uncanceled numbers in the divisor. The result will be the 
quotient. The product . of all the uncanceled numbers in 
the dividend equals 2x1x4x1 = 8; the product of all the 
uncanceled numbers in the divisor equals lx lXlXl=l. 

„ 2X1X4X18 oA 

Hence, = - = 8. Ans. 

1 X 1 X 1 X 1 1 

2 

pp x ;#- x % x i 
p I l 
l i 



10 ARITHMETIC. 

(b) (80 X 60 X 50 X 16 X 14) + (70 X 50 X 24 X 20). 
Placing the numerator over the denominator, the problem 

becomes 

80 X 60 X 50 X 16 X 14 _ ? 
70 X 50 X 24 X 20 

The 50 in the dividend and 70 in the divisor are both divis- 
ible by 10, since 50 divided by 10 equals 5, and 70 divided 
by 10 equals 7. Cross off the 50 and write the 5 over it; 
also, cross off the 70 and write the 7 under it. Thus, 

5 
80 X 60 X ^ X 16 X 14 

;p X 50 x 24 X 20 

7 

The 50 and 70 are not to be considered any longer, and 5 
in fact, may be erased entirely and the 5 and 7 placed in 
their stead, and treated as if the 50 and 70 never existed. 
Thus, 

80 X 60 X 5 X 16 X 14 _ 
7 X 50 X 24 X 20 ~ 

Also, 80 in the dividend and 20 in the divisor are divisible 
by 20, since 80 divided by 20 equals 4, and 20 divided by 20 
equals 1. Cross off the 80 and write the 4 over it; also, 
cross off the 20 and write the 1 under it. Thus, 

4 

M X 60 X 5 X 16 X 14 



7 X 50 X 24 X ?jJ 

1 



The 80 and 20 are not to be considered any longer, and, 
in fact, may be erased entirely and the 4 and 1 placed in 
their stead, and treated as if the 80 and 20 never existed. 
Thus, 

4 X 60 X 5 X 16 X 14 _ 
7 X 50 X 24 X 1 

Again, 16 in the dividend and 24 in the divisor are divisible 
by 8, since 16 divided by 8 equals 2, and 24 divided by 8 
equals 3. Cross off the 16 and write the 2 over it; also cross 
off the 24 and write the 3 under it. Thus, 



ARITHMETIC. 11 

2 
4 x 60 x 5 x 1$ X 14 _ 

7 X 50 x #£ X 1 " 
3 

The 16 and 24 are not to be considered any longer, and, 
in fact, may be erased entirely and the 2 and 3 placed in 
their stead, and treated as if the 16 and 24 never existed. 
Thus, 

4X60X5X2X14 ^ 
7 X 50 X 3 X 1 

Again, 60 in the dividend and 50 in the divisor are divis- 
ible by 10, since 60 divided by 10 equals 6, and 50 divided by 
10 equals 5. Cross off the 60 and write the 6 over it; also, 
cross off the 50 and write the 5 under it. Thus, 

6 

4x00X5x2xl4 = 

7X^X3X1 
5 

The 60 and 50 are not to be considered any longer, and, in 
fact, may be erased entirely and the 6 and 5 placed in their 
stead, and treated as if the 60 and 50 never existed. Thus, 

4X6X5X2X14 ^ 
7X5X3X1 " 

The 14 in the dividend and 7 in the divisor are divisible by 
7, since 14 divided by 7 equals 2, and 7 divided by 7 equals 1. 
Cross off the 14 and write the 2 over it; also, cross off the 7 
and write the 1 under it. Thus, 

2 
4x6x5x2x^ = 
Jx5x3xl 
1 

The 14 and 7 are not to be considered any longer, and, in 
fact, may be erased entirely and the 2 and 1 placed in their 
stead, and treated as if the 14 and 7 never existed. Thus, 

4X6X5X2X2 _ 
1X5X3X1. ~~ . 
G. G. IV.— 2 



12 ARITHMETIC. 

The 5 in the dividend and 5 in the divisor are divis- 
ible by 5, since 5 divided by 5 equals 1. Cross off the 5 
of the dividend and. write the 1 over it; also, cross off the 5 
of the divisor and write the 1 under it. Thus, 

1 
4x6xfiX 2x2 = 

1 X £ X 3 X 1 
1 

The 5 in the dividend and 5 in the divisor are not to be 
considered any longer, and, in fact, may be erased entirely 
and 1 and 1 placed in their stead, and treated as if the 5 and 
5 never existed. Thus, 

4X 6 X 1 X 2X 2 _ 
1X1X3X1 ~~ 

The 6 in the dividend and 3 in the divisor are divisible by 
3, since 6 divided by 3 equals 2, and 3 divided by 3 equals 1. 
Cross off the 6 and place 2 over it ; also, cross off the 3 and 
place 1 under it. Thus, 

2 
4x 0X1 X 2x2 _ 

lxlx?xl ~ 
1 

The 6 and 3 are not to be considered any longer, and, in 
fact, may be erased entirely and 2 and 1 placed in their 
stead, and treated as if the 6 and 3 never existed. Thus, 
4x2x1x2x2^32 
lxlxlxl 1 
2 1 
4 2 2 
?0X(3PxgPx;gX# _ 4x2xlx2x2 _32_ qo 

ce ' 7Pxspx2£x2p " lxlxlxl ~ 1 ~ 

I } £ . 1 Ans - 

1 11 

(19) 28 acres of land at $133 an acre would cost 

28 X $133 = 83,724, 



1064 
266 
$3724 



ARITHMETIC. 13 

If a mechanic earns $1,500 a year and his expenses are 
$968 per year, then he would save $1500—1968, or $532 
per year. 9 6 8 

$532 

If he saves $532 in 1 year, to save $3,724 it would take as 
many years as $532 is contained times in $3,724, or 7 years. 

532)3724(7 years. Ans. 
3724 



(20) If the freight train ran 365 miles in one week, and 
3 times as far lacking 246 miles the next week, then it ran 
(3 X 365 miles) — 246 miles, or 849 miles the second week. 
Thus, 3 6 5 

3 



1095 
246 



difference 8 4 9 miles. Ans. 

(21) The distance from Philadelphia to Pittsburg is 354 
miles. Since there are 5,280 feet in one mile, in 354 miles 
there are 35* X 5,280 feet, or 1,869,120 feet. If the driving 
wheel of the locomotive is 16 feet in circumference, then in 
going from Philadelphia to Pittsburg, a distance of 1,869,- 
120 feet, it will make 1,869,120 -r- 16, or 116,820 revolutions. 

16)1869120(116820 rev. Ans. 
16 

~6 
16 

109 

96 

131 

128 

~32 
32 



14 



(22) 



ARITHMETIC. 

(a) 576)589824(1 02 4 Ans. 
576 
1382 
1152 



2304 
2304 



V) 



43911) 369730620 (8420 Ans. 
351288 

184426 
175644 



87822 
87822 



(c) 





505) 2527525 ( 5005 Ans. 
2525 



2525 
2525 



(d) 1234)4961794302(4020903 Ans, 
4936 



2579 
2468 

11143 
11106 



3702 
3702 

(23) The harness evidently cost the difference between 
$444 and the amount which he paid for the horse and wagon. 

Since $264+1153 = $417, the amount paid for the horse 
and wagon, $444 — $417 = $27, the cost of the harness. 

$264 $444 

153 417 

$ 4 1 7 $27 Ans. 



ARITHMETIC. 15 



(24) (a) 1024 

576 



6144 
7168 
5120 

5 8 9 8 2 4 Ans. 



(J) 5 5 

505 



25025 
2 5 2 5 

2527525 Ans. 



(r?) 43911 

8420 



878220 
175644 
351288 

369730620 Ans. 

(25) Since there are 12 months in a year, the number of 
days the man works is 25 X 12 = 300 days. As he works 10 
hours each day, the number of hours that he works in one 
year is 300 X 10 = 3,000 hours. Hence, he receives for his 
work 3,000 X 30 = 90,000 cents, or 90,000 ~ 100 = $900. Ans. 

(26) See Art. 71. 

(27) See Art. 77. 

(28) See Art. 73. 

(29) See Art. 73. 

(30) See Art. 75. 

13 

(31) -3- is an improper fraction, since its numerator 13 

8 

is greater than its denominator 8. 

(32) 4;i4> 8 4 



16 ARITHMETIC. 

(33) -To reduce a fraction to its lowest terms means to 
change its form without changing its value. In order to do 
this, we must divide both numerator and denominator by 
the same number until we can no Jonger find any num- 
ber (except 1) which will divide both of these terms without 
a remainder. 

4 
To reduce the fraction — to its lowest terms we divide 
8 

both numerator and denominator by 4, and obtain as a 

1 4—41 4—4 
result the fraction — . Thus, - . = — ; similarly, — ' = 

2 8^4 2 ^16^4 

1_. _^-^-4: = 2 -r-2 = l . 32-T-8 _4 -^-4 = 1 

4 ; 32^-4~8-^2 4 ' 64-^-8 ~~ 8 -f4~2' nS * 

(34) When the denominator of any number is not 
expressed, it is understood to be 1, so that — is the same as 

6-^-1, or 6. To reduce - to an improper fraction whose 

denominator is 4, we must multiply both numerator and 
denominator by some number which will make the denomi- 
nator of 6 equal to 4. Since this denominator is 1, by mul- 
tiplying both terms of — by 4 we shall have — . = — . 
r j & x j 1X44' 

which has the same value as 6, but has a different form. Ans. 

(35) In order to reduce a mixed number to an improper 
fraction, we must multiply the whole number by the denom- 
inator of the fraction and add the numerator of the fraction 
to that product. This result is the numerator of the improper 
fraction, of which the denominator is the denominator of the 
fractional part of the mixed number. 

7 7 8 

7— means the same as 7 -j- — . In 1 there are—, hence in 

8 8 8 

7 there are 7 X 5- = -5- ; -5- plus the — of the mixed number 
8 8 8 8 

= — - -f- q — -3-, which is the required improper fraction. 

8 8 8 

5 (13 X 16) + 5 _ 213 3 _ (10 X 4) + 3 43 
16 16 16 ' 4 4 4* 



ARITHMETIC. 17 

(36) The value of a fraction is obtained by dividing the 

numerator by the denominator. 

13 
To obtain the value of the fraction — we divide the num- 

Z 

erator 13 by the denominator 2. 2 is contained in 13 six 

times, with 1 remaining. This 1 remaining is written over 

the denominator 2, thereby making the fraction — , which is 

/© 

annexed to the whole number 6, and we obtain 6— as the 

Z 

mixed number. The reason for performing this operation is 

2 13 

the following: In 1 there are — (two halves), and in — (thir- 

Z Z 

teen halves) there are as many units (1) as 2 is contained 

times in 13, which is 6, and — (one-half) unit remaining. 

Z 

13 11 

Hence, — = 6 + — = 6-, the required mixed number. Ans. 
±Z z z 

17 A 1 A 69 A 5 A 16 o A 67 

T = 4 4- AnS - 16 = % AnS - -8= % - AnS - 64 = 
, 3 

lei- Ans - 

(37) In division of fractions, invert the divisor (or, in 
other words, turn it upside down) and proceed as in multi- 
plication. 



M 35 . 5 _35 vr 16_ 35 X 16 _ 560 

W 35 ' 16 1 X 5 1X5 5 112 - 


Ans. 


(b \ 9 . o _ 9 . 3 9 1 9X1 9 

K } 16 ' 16 * 1 16 X 3 16 X 3 ~48~ 


3 . 
= 1? AnS 


( c \ 17 . a _ 17 . 9 17 v 1 17 X 1 17 
K } 2 2 ' 1 2 X 9 : 2 X 9 ~ 18* 


Ans. 


(d) 113 • 7 _ 113 v, i6 H3X16 1,808 
K } 64 * 16 64 7 " 64 X 7 448 
3 


_452 
" 112 " 


8)118(41. Ans. 
3 12 28 





18 ARITHMETIC. 

3 3 

(e) 15— -4- 4— = ? Before proceeding with the division, 

4 8 

reduce both of the mixed numbers to improper fractions. 
Thus 15 3_ (15x4) + 3 _60 + 3_63 3_ (4 X 8) + 3 _ 

i nus, i5 4 - 4 - 4 - 4 > ana4 8 - 8 - 

32 + 3 35 _„ , . . 63 35 A , £ 

— - 1 — = — - . The problem is now — -s- — = ? As before, 

8 8 4 8 

■ ,. . -, u . . 63 35 63^ 8 63 X 8 

invert the divisor and multiply ; -—-=- — = — x 



4 8 4 35 4X 35 
504 _ 252 _ 126 _ 18 
140 ~ 70 "~ 35 "" 5' 

^ 3 

5)18(3^ Ans - 

1_5 5 
3 

When the denominators of the fractions to be added dr* 
alike, we know that the units are divided into the same 
number of parts (in this case eighths) ; we, therefore, add the 
numerators of the fractions to find the number of parts 

Q 

(eighths) taken or considered, thereby obtaining — or 1 as 

o 

the sum. 

(39) When the denominators are not alike we know that 
the units are divided into unequal parts, so before adding 
them we must find a common denominator for the denom- 
inators of all the fractions. Reduce the fractions to fractions 
having this common denominator, add the numerators and 
write the sum over the common denominator. 

In this case, the least common denominator, or the least 
number that will contain all the denominators, is 16; hence, 
we must reduce all these fractions to sixteenths and then add 
their numerators. 

13 5 1 

j + 7T- + — = ? To reduce the fraction - to a fraction 

having 16 for a denominator, we must multiply both terms 



ARITHMETIC. 19 

of the fraction by some number which will make the denom- 

1x4 4 
inator 16. This number evidently is 4, hence, — = — . 

4X4: J_0 

Similarly, both terms of the fraction — must be multiplied 

3X2 6 
by 2 to make the denominator 1G, and we have - =77- 

o X <* It) 

The fractions now have a common denominator 1G; hence, 

we find their sum by adding the numerators and placing their 

sum over the common denominator, thus: — -f- — -f- — = 

lb Id lb 

4+6 + 5 = 15 Ans _ 
16 16 



(40) When mixed numbers and whole numbers are to be 
added, add the fractional parts of the mixed numbers sep- 
arately, and if the resulting fraction is an improper fraction, 
reduce it to a whole or mixed number. Next, add all the 
whole numbers, including the one obtained from the addition 
of the fractional parts, and annex to their sum the fraction 
of the mixed number obtained from reducing the improper 
fraction. 

5 7 5 

42 + 31- -f- 9— = ? Reducing - to a fraction having 

a denominator of 16, we have — X n = Ta- Adding the two 

o Z lb 

10 7 
fractional parts of the mixed numbers we have - — f- — = 
F 16~16 

10 + 7 _ 17 1 

16 ~16~ 16* 

The problem now becomes 42 -f- 31 + 9 + 1— = ? 

Adding all the whole numbers and the 



number obtained from adding the fractional 

li parts of the mixed numbers, we obtain 83— 

1 6 16 

83^ Ans. as their sum. 



20 ARITHMETIC. 

(41) 2 9| + 50| + 41 +69 A= ? |=j;{ = §. 

5_ 5 X 2_10 12 , 10 _3 _ 12 + 10 -f 3 _ 25 _ _9 
8 " 8 X 2 " 16' 16 + 16 + 16 ~~ 16 ~ 16 16 

9 
The problem now becomes 29 + 50 + 41 + 69 + 1— = ? 



16 



29 square inches. 
50 square inches. 
41 square inches. 
69 square inches. 
1 T 9 ¥ square inches. 



190 T 9 T square inches. Ans. 

16 

3 
The line between 7 and — means that 7 is to be divided 

16 



y 16 



15 3 



w _5 ~ 32 ' 8 ~ 32 X 5 ~ ffi x £ ~ 4' 
8 4 

4+3 7 
^^F = !=8^=ro- (See Art. 131.) An, 

7 

(43) -3 = value of the fraction, and 28 = the numerator, 

o 

We find that 4 multiplied by 7 = 28, so multiplying 8, the 
denominator of the fraction, by 4, we have 32 for the required 

28 7 
denominator, and — = — . Hence, 32 is the required de- 
oZ o 

nominator. Ans. 

7 7 

(44) (a) - — — = ? When the denominators of frac- 

8 16 

tions are not alike it is evident that the units are divided 
into unequal parts, therefore, before subtracting, reduce the 



ARITHMETIC. 21 

fractions to fractions having a common denominator. Then, 
subtract the numerators^ and place the remainder over the 
common denominator. 

^7x2_14 14 7 = 14-7 _ 7 . 

8X2~16' 16 16" 1G 16* nS * 

7 
(£) 13 — 7— = ? This problem may be solved in two 

ways: 

/7TJ/.- 13 = 12—, since — = 1, and 12— = 12 -f- — = 

32 + 1 = 13. 

12|f We can now subtract the whole numbers sepa- 

7 T 7 ¥ rately, and the fractions separately, and obtain 12 — 7 

T7 - .16 7 16-7 9 „ , 9 9 A 

^ = 5and---=-^- = r6 . 5 + I - 6 = 5-. Ans. 

Second : By reducing both numbers to improper fractions 

having a denominator of 16. 

13 = J_3 X 16 208 _7 __ (7 X 16) + 7 _ 112 + 7 

1 1X16 16' 16 16 16 

119 

16* 

^ '. , 208 119 208-119 89 - 

Subtracting, we have — — r = • — = — ; and 

lb lb lb lb 

89 _ im a/ k the same result that was obtained by the 
16 - 16 )°9(5w first method . 
80 

~9 (c) 312-^ - 229^- = ? We first reduce 

— lb 3% 

16 the fractions of the two mixed numbers to 

fractions having a common denominator. Doing this we 

, 9 9 X 2 18 TTr - , - 

nave — = — — = — - We can now subtract the whole 

lb lb X 2 6Z 

numbers and fractions separately, and have 312 — 229 = 83 
and L8_A = 18-o 13 



32* 

1 8 
3T 



312J 



229- 5 - < • I 3 — I 3 



88H 



^ 83 + 32 = 83 32' AnS ' 



22 ARITHMETIC. 

(45) The man evidently traveled 85 — + 78 -^ + 125 ^ 

X/i J.0 oD 

miles. 

Adding the fractions separately in this case, 
5 9 17_ 5 3 17_ 175 + 252 + 204 _631_ 211 
12 + 15 + 35 _ 12 + 5 + 35 - 420 ~ 420 420' 

Adding the whole numbers and the mixed number 85 

representing the sum of the fractions, the sum is 78 

289 gi miles. Ans. **J^ 

To find the least common denominator, we have ogoTTT 

5 )12, 5, 35 

7 )12, 1, 7 

12, 1, 1, or 5X7X12 = 420. 

(46) 573|ton, *=» 
216 1 ton, ! = » 



7 7 

difference 357 — tons. Ans. — = difference. 

•^40 40 M 

(47) Reducing 9 — to an improper fraction, it becomes 

37 -_ u . . . 37, 3 37 3 111 _ 15 , n A 

■j-. Multiplying _ by -, — X g = ^- = 3 — dollars. Ans 

(48) Referring to Arts. 114 and 116, 

— of — of — of — of 11 multiplied by — of — of 45 = 
o 4 11 20 o o 

3 

V 

?x£x7x!9x;ix7x5x / 4g _ 7x19x7x5x3 13,965 

^X4x;;x2pxlx8x^xl 4x4x8 128 

4 fi 

1 3 
109 _. An, 

4 6 

(49) |ofl6=|x^ = 12. 12-r-| = ^x| = 18. Ans. 

1 7 845 15 

(50) 211 j X 1 q- = —j— X -5-, reducing the mixed numbers 

* o .4 o 



ARITHMETIC. 23 

, ,. 845 15 12,675 

to improper fractions. — r- X -77 = — b: — cents = amount 

4 8 32 

paid for the lead. The number of pounds sold is evidently 

2,535 

12,675 1 tf,ffl&$ 2,535 1KQ 7 , ^ u 

-32~ + 2 2 = ^^ x J = TT = 158 I6 pounds ' The 

16 

• • -Oil 1 1«Q 7 845 2 ' 535 3 ' 380 

amount remaining is 211 158 



4 16 4 16 16 



2,535 845 ^ 13 , . 

-Tr = i6- = 52 r6 pounds - Ans - 



£ £ 
a s 

(51) 8 = £y^/ hundredths. 







to u 

ago 
^ ^ 5 
13 1 = C«* hundred thirty-one thousandths. 



3*2 3~ 
a 2 o a 

CO S A <tt 

1 = 0«* ten-thousandth. 



« t § 2 g S 

a 5 o a § 3 

2 7= Twenty-seven millionths. 



il 



n 3 o a 



10 8 = <?#* hundred eight ten-thousandths. 



24 ARITHMETIC. 



25 s 

■s Is 



H i-i « l-c 

93.0 10 1 = Ninety-three, and one hundred one ten°thousandths 

In reading- decimals, read the number just as you would ii 
there were no ciphers before it. Then count from the decimal 
point towards the right, beginning with tenths, to as many 
places as there are figures, and the name of the last figure 
must be annexed to the previous reading of the figures to 
give the decimal reading. Thus, in the first example above, 
the simple reading of the figure is eight, and the name of its 
position in the decimal scale is hundredths, so that the 
decimal reading is eight hundredths. Similarly, the fig- 
ures in the fourth example are ordinarily read twenty-seven ; 
the name of the position of the figure 7 in the decimal scale 
is millionths, giving, therefore, the decimal reading as 
twenty-seven millionths. 

If there should be a whole number before the decimal 
point, read it as you would read any whole number, and 
read the decimal as you would if the whole number were 
not there; or, read the whole number and then say, " and " 
so many hundredths, thousandths, or whatever it may be, 
as "ninety-three, and one hundred one ten thousandths." 

(52) See Art. 139. 

(53) See Art. 153. 

(54) See Art. 160. 

(55) A fraction is one or more of -the equal parts of a 
unit, and is expressed by a numerator and a denominator, 
while a decimal fraction is a number of tenths, hundredths, 
thousandths, etc., of a unit, and is expressed by placing a 
period (.), called a decimal point, to the left of the figures 
of the number, and omitting the denominator. 

(56) See Art. 165. 



ARITHMETIC. 25 

(57) To reduce the fraction -to a decimal, we annex 

one cipher to the numerator, which makes it 1.0. Dividing 
1.0, the numerator, by 2, the denominator, gives a quotient 
of .5, the decimal point being placed before the one figure 
of the quotient, or .5, since only one cipher was annexed to 
the numerator. Ans. 

1 A 

8 )7.000 32) 5.00000 (.15 62 5 Ans. 

.8 7 5 Ans. 3 2 



c- r* 65 ±u 65 18 ° 

Smce.G5 = — ,then,_ ^ 

must equal .65. Or, when oqa 105 

the denominator is 10, 100, 192 1000 ~ * 125, AnS * 

1000, etc., point off as many 

places in the numerator as 

there are ciphers in the _ 

denominator. Doing so, 160 

65 =.65. An, 122 



100 

(58) (a) This example, written in the form of a fraction, 
means that the numerator (32.5 -f- .29 + 1.5) is to be divided 
by the denominator (4.7 + 9). The operation is as follows: 

32.5 + .29 + 1.5 



— ? 



47 + 9 

3 2.5 
+ .29 
+ 1.5 



4.7 689 

9.0 685 



13.7 ) 3 4.2 9 000 ( 2.5 02 9 Ans. 

Since there are 5 deci- 
mal places in the dividend 
and 1 in the divisor, there 
13.7 40 are 5 — 1 or 4 places to 

2 7 4 be pointed off in the quo- 

12 6 tient. The fifth figure of 
12 3 3 the decimal is evidently 
2 7 less than 0. 



2G 



ARITHMETIC. 



{b) Here again the problem is to divide the numerator, 
which is (1.283 X 8 -f- 5), by the denominator, which is 2.63. 
The operation is as follows: 



1.283 X 8 + 5 
2.63 


= ? 8 + 5 = 13. 
1.283 






2.6 3 


X 13 

3849 
1283 
)16.679000( 

1578 


6.3418 



Ans. 



899 
789 
1100 
1052 
480 



480 
263 
2170 
2104 
66 



589 + 27 X 163 - 8 
\ c ) 25 -4- 39 



25 + 39 

589 
+ 27 

6l~6 



25 

+ 39 

64 



163 

- 8 

155 
X616 
930 
155 
930 



64)95480.000(1491.875 



64 

314 

256 
588 
576 



Ans. 



There are three deci- 
mal places in the quotient, 
since three ciphers were 
annexed to the dividend. 



120 

64 

560 

512 



480 

448 
320 
320 



ARITHMETIC. 27 



. 40.6 + 7.1 X (3.029-1.874) 
{ ' 6.27 + 8.53-8.01 

40.6 3.0 2 9 

+ 7.1 - 1.87 4 



47.7 1.15 5 

X 47.7 

6.2 7 

8.5 3 



8 08 5 
80 85 



14.80 4620 

- 8.01 



6.79)5 5.0 9 3 500 ( 8.113 9. Ans. 



6.79 5432 



945 
679 



6 decimal places in 6 79 

the dividend — 2 deci- 
mal places in the divi- 
sor = 4 decimal places 
to be pointed off in 2 6 60 

the quotient. 2037 

6230 
6111 



119 



< 59 > • 875 = W0 = S = -8 ofafoOt - 

1 foot = 12 inches. 

3 

| of 1 foot = lx^ = ~ = 10l inches. Ans. 
o p 1 A A 

2 
(60) 12 inches = 1 foot. 

1 of an inch = 1-12 = 1x^ = 6^ of a foot. 

4 
Point off 6 decimal places in the quotient, since we 
annexed six ciphers to the dividend, the divisor con- 
taining no decimal places; hence, 6 — 0=6 places to be 
pointed off. 

(/. a. IV.— 8 



ARITHMETIC. 

64 ) 1.000000(. 015625 Ans. 
64 



360 
320 

400 
384 


160 
128 



320 
320 



(61) If 1 cubic inch of water weighs .03617 of a pound, 
the weight of 1,500 cubic inches will be .03617 X 1,500 = 
54255 lb. 

.03617 lb. 
1500 



1808500 
3617 



54.2 5 5 00 lb. Ans. 
(62) 72. 6 feet of fencing at $. 50 a foot would cost 

7 2.6 X .50, or $36.30. 
.50 



$3 6,300 



If. by selling a carload of coal at a profit of $1.65 per ton s 
I make $36.30, then there must be as many tons of coal in 
the car a* 1.65 is contained times in 36.30, or 22 tons. 

1.65 ) 3 6.3 ( 22 tons. Ans. 
330 

330 
330 



ARITHMETIC. 29 

(63) £31 ) 17892.00000 ( 77.45454, or 77.4545 tc 
16 17 four decimal places. Ans 

17 22 
1617 



1050 
924 



1260 
1155 

1050 
924 

1260 
1155 

1050 

37.13 % .0952 

, , n.w x % x am x 19 x 19 x 350 _ 
1,000 % 

37.13 x .0952 x 19 X 19 x 350 _ 446,618.947600 
1,000 
446.619 to three decimal places. Ans. 

37.13 19 361 

.0952 19 350 



7426 171 18050 

18565 19 1083 

33417 ^Tj" 126350 



3.534776 



1,000 


3.534776 


126350 


176738800 


10604328 


21208656 


7069552 


3534776 


446618.947600 



(65) See Art. 174. Applying rule in Art. 1 75, 

/ \ »„»„ 64 50.7392 51 . 
(a) .7938 x- 4 = -^ 

W -47915 xJ-J = 



64 


4.5312 


32 


7.6664 



32* 


Ans. 


8 
16 : 


= 2' 



30 ARITHMETIC. 

(66) In subtraction of decimals, /^ 7 9 6 3 
place the decimal points directly 8 514 

under each other, and proceed as in 

the subtraction of whole numbers, 7 8.7 7 8 6 Ans. 

placing the decimal point in the remainder directly under 
the decimal points above. • , 

In the above example we proceed as follows : We can not 
subtract 4 ten-thousandths from ten-thousandths, and, as 
there are no thousandths, we take 1 hundredth from the three 
hundredths. 1 hundredth = 10 thousandths = 100 ten-thou- 
sandths. 4 ten-thousandths from 100 ten-thousandths leaves 
96 ten-thousandths. 96 ten-thousandths = 9 thousandths -j- 6 
ten-thousandths. Write the 6 ten-thousandths in the ten- 
thousandths place in the remainder. The next figure in the 
subtrahend is 1 thousandth. This must be subtracted from 
the 9 thousandths which is a part of the 1 hundredth taken 
previously from the 3 hundredths. Subtracting, we have 1 
thousandth from 9 thousandths leaves 8 thousandths, the 8 
being written in its place in the remainder. Next we have 
to subtract 5 hundredths from 2 hundredths (1 hundredth 
having been taken from the 3 hundredths makes it but 2 
hundredths now). Since we can not do this, we take 1 tenth 
from 6 tenths. 1 tenth ( = 10 hundredths) -f- 2 hundredths 
= 12 hundredths. 5 hundredths from 12 hundredths leaves 
7 hundredths. Write the 7 in the hundredths place in the 
remainder. Next we have to subtract 8 tenths from 5 tenths 
(5 tenths now, because 1 tenth was taken from the 6 tenths). 
Since this can not be done, we take 1 unit from the 9 units. 
1 unit = 10 tenths ; 10 tenths -f- 5 tenths = 15 tenths, and 8 
tenths from 15 tenths leaves 7 tenths. Write the 7 in the 
tenths place in the remainder. In the minuend we now have 
708 units (one unit having been taken away) and units in the 
subtrahend. units from 708 units leaves 708 units; hence, 
we write 708 in the remainder. 

(b) 81.9 6 3 (c) 18.00 (d) 1.000 

1.700 .18 .001 



80.2 6 3 Ans. 17.8 2 Ans. .9 9 9 Ans. 



ARITHMETIC. 31 

(e) 872.1 -(.8721 + .008)=? Inthisprob- 

iem we are to subtract (.8721 + .008) from - 8 ? 21 

8 
872.1. First perform the operation as indi- ______ 



5.0280 



cated by the sign between the decimals .8801 sum. 
enclosed by the parenthesis. 

Subtracting the sum (obtained by adding the decimals 

8 7 2 10 enclosed within the parenthesis) from 

g g i the number 872.1 (as required by the 

minus sign before the parenthesis), 

871 91 QQ An? 

o i j.,/0 j. j j xiaib. we b ta j n the required remainder. 

(/) (5.028 + .0073) - (6.704 - 2.38) = ? First perform 
the operations as indicated by the signs be- 
tween the numbers enclosed by the paren- 
theses. The first parenthesis shows that ' 

5.028 and .0073 are to be added. This 5.03 53 sum. 
gives 5.0353 as their sum. 

The second parenthesis shows that 

2 - 380 2.38 is to be subtracted from 6.704. 

4.3 2 4 difference. The difference is found to be 4.324. 

The sign between the parentheses indicates that the 

g q 3 5 3 quantities obtained by performing 

4 3 2 4 tne a bove operations, are to be sub- 

tracted, namely, that 4.324 is to be 

./±±d _ms. subtracted from 5.0353. Perform- 

ing this operation we obtain .7113 as the final result. 

(67) In subtracting a decimal from a fraction, or sub- 
tracting a fraction from a decimal, either reduce the fraction 
to a decimal before subtracting, or reduce the decimal to a 
fraction and then subtract. 

7 7 

(a) — — .807 = ? — reduced to a decimal becomes 

o 5 

7 
8 )7.000 

.875 

.8 7 5 

* 07 Subtracting .807 from .875 the re- 

mainder is .068, as shown. 



.0 6 8 Ans. 



32 ARITHMETIC. 



(6) .875 — -= ? Reducing .875 to a fraction we have 

Q „._ 875 _175_35_7 7 3_7-3 4 1 

b ' O -l,000-200-40-8 ; henCe, 8~8-~T~ = 8 = 2"' 
3 o Ans. 

Or, by reducing -to a decimal, -^ Q00 and then sub- 



tracting;, we obtain .875 - .375 = .5 = — = - 875 

.3 7 5 
— , the same answer as above. ~5~00 Ans 



^ (iL + ' 435 ) ~ (wd ~ - 07 ) = ? We first Perform the 
operations as indicated by the signs between the numbers 
enclosed by the parentheses. Reduce — to a decimal and 



5 



32 



we obtain — = .15625 (see example 57). 

21 
Adding .15625 and .435, .15 6 25 -,— = .21; subtracting, .21 

.435 .07 



sum .5 9125 differ e7ice .14 

We are now prepared to perform the .59125 
operation indicated by the minus sign be- -14 
tween the parentheses, which is, difference m 4 5 1 2 5 Ans. 
(d) This problem means that 33 millionths and 17 thou- 
sandths are to be added. Also, that 53 hundredths and 
274 thousandths are to be added, and the smaller of these 
sums is to be subtracted from the larger sum. Thus, 
(.53 + .274) - (.000033 + .017) = ? 



3 
I $ « .c 5 .8 04 larger sum, 

2 g I "g I ? « .017033 smaller sum. 



-H S «A 



^ c o a «S ego difference .786967 Ans. 

.0 00033 .5 3" 

.0 17 .2 7 4 



.017033 sum. .804 sum. 



ARITHMETIC. 33 



(68) In addition of decimals the 
decimal points must l/e placed directly 
under each other, so that tenths will 
come under tenths, hundredths under 
hundredths, thousandths under thou- 
sandths, etc. The addition is then 
performed as in whole numbers, the 
decimal point of the sum being placed 
directly under the decimal points above. 



(69) 9 2 7.416 (70) 

8.274 
3 7 2.6 
62.0 7 9 38 



.7 




.089 




.4005 




.9 




.000027 




2.21452 7 


Ans. 


co 




& 




-u 




T3J 




c/i G 




•^ m 




-u 22 




. T3 3 




1 a aj 


CO 


S-2 1.3 


J3 
-(-> 


tenths, 
hundre 
thousai 
ten-tho 
hundre 


1 


.017 




.2 




.000047 





(71) («) 

.107 
.013 


321 

107 


.001391 


(£) 2 3 
2.0 3 


609 
406 


412.0 9 
.203 


1236 27 
824180 


83.65427 



1370.3 6 9 38 Ans. 



.217047 = Ans. 

There are 3 decimal places in the multi- 
plicand and 3 in the multiplier; hence, 
there are 3 -J- 3 or 6 decimal places in 
the product. Since the product con- 
tains but four figures, we prefix two 
ciphers in order to obtain the neces= 

ns * sary six decimal places. 

There are two decimal places in the 
multiplier and none in the multipli- 
cand; hence, there are 2 -j- or two 
decimal places in the first product. 

Since there are 2 decimal places in 
the multiplicand and 3 decimal places 
in the multiplier, there are 3 -j- 2 or 5 
decimal places in the second product 

Ans 



34 ARITHMETIC. 

(c) First perform the operations indicated by the signs 
between the numbers enclosed by the parenthesis, and then 
perform whatever may be required by the sign before the 
parenthesis. 

Multiply together the numbers 2.7 3 1.8 5 

and 31.85. 2.7 

The parenthesis shows that .316 is 2 2 2 9 5 

to be taken from 3.16. 3.160 6370 

.316 



85.995 



2.844 
The product obtained by the first 
operation is now multiplied by the 8 5.9 9 5 

remainder obtained by performing 2.8 4 4 

the operation indicated by the signs 
within the parenthesis. 

687960 
171990 



244.569780 Ans. 



(d) (107.8 + 6.541 - 31.96) X 1.742 = ? 

107.8 
+ 6.541 

114.341 
- 31.96 



8 2.3 81 
X 1.742 

164762 
329524 
576667 
82381 



14 3.50 7 7 02 Ans. 



(72) (a) (1-.13JX. 625 + |=? 

First perform the operation indicated by the parenthesis. 



ARITHMETIC. 35 

L--1 

16 "" 16) 7.0000 (.4 3 7 5 We point off four decimal 

6 4 places since we annexed four 

~T7j ciphers. 

48 



120 
112 



. 80 
80 

.4375 
.13 

Subtracting, we obtain .3075 

The vinculum has the same meaning as the parenthesis; 
5 _ 5 hence, we perform the operation indicated 

8 "~ 8 ) 5.0 by it. We point off three decimal places, 
.6 2 5 since three ciphers were annexed to the 5. 

Adding the terms in- .6 2 5 
eluded by the vinculum, .6 2 5 
we obtain 1.2 5 

The final operation is to perform the work indicated by 
the sign between the parenthesis and the vinculum, thus, 

.3075 
1.25 









15375 


















6150 


















3075 


Ans. 












.384375 




(*) 


A- 9 

\32 


X .2l|- 


(°*>4) = 


: ? 










.21 = 


21 
100' 


19 21 
32 100 


- 3 " 02- 
3200' '° Z - 


2 

z ioo' 


2 

10C 


,4= 


6 

1600 


_ 3 

800' 


3 


3 


X 4 


12 399 


12 




399- : 


12 


387 


800" 


" 800 X 4 ~ 3200' 3200 


3200 




3200 




3200 



36 ARITHMETIC. 

387 
Reducing to a decimal, we obtain 

O/dUO 

387 

3200 ) 38 7.0 000000 (.1209375 Ans. 
3200 



6700 
6400 


30000 

28800 


12 

9600 


24000 
22400 


16000 
16000 



Point off seven decimal 
places, since seven ciphers 
were annexed to the divi- 
dend. 



to 


ff 


+ 


.013 


-2.17) 


X 13 T 
4 


-7^ 
16 


— } 






13 


13 






Point 


off 


two 


decimal 




3.25 


4 


" 4) 


13.00 


places, 


since 


two 


ciphers 


+ 


.013 



3.2 5 were annexed to the divi- 



3.2 6 3 

5 dend ' -2.17 

— reduced to a decimal is .3125, since 

_5 

16)5.0000(.3125 Point off four decimal 

4 8 places, since four ciphers 

2 were annexed to the 

1 6 dividend. 



40 
32 

80 
80 

Then, 7 ^ = 7. 3125, and 13y = 13. 25, since \ = \ . , 

16 4 4 4) 1.00 

~ 25 



ARITHMETIC. 37 

13.2 5 5.93 7 5 

- 7.3125 X 1.093 

5.9375 178125 

534375 
593750 
6.48 9 6 87 5 Ans. 

(73) (a) .875 H-i =.875 -S-.5 (since 4 j=.o) = 1.75. Ans. 

Another way of solving this is to reduce .875 to its equivalent 
common fraction and then divide. 

q»* ? • ovk 875 175 35 7 + , 7 

.875 = -, since. 875 = — = _ = _ = _; then, - + 

1_7 v ?_7__ -3 «. 3_3 i 3 _i7^ 

2~? X 1~4~ 1 4* &mCe 4 ~4)3.00(.75, *! ~ lm I0 ' 

* 28 

the sameanswer as above. 2 

20 



W 8^- 5 = 8^2( SmCe ' 5 = 2)^F X l 



? 7 .3 
4 4 



1.75. Ans. 4 

7 
This can also be solved by reducing — to its equivalent 

o 

decimal and dividing by .5;-| = .875; .875 -f-.. 5 = 1.75. 

8 

Since there are three decimal places in the dividend and one 
in the divisor, there are 3 — 1, or 2 decimal places in the 
quotient. 

, . .375 X i _ We shall solve this problem by first 

-rw — -125 ~~ reducing the decimals to their equiva- 
lent common fractions. 
375 75 15 3 3 1 3 . . , 

• 375 = 1^00 = 200 = 40 = 8' sX 1 = ^,or the value of 

the numerator of the fraction. 

125 25 1 1 

■ 125 = 1^00 = 200 = 8- Reducin S 8 t0 sixteenths > we 

have I x 2 = re- Then - h - re = h' or the value of the de - 



38 ARITHMETIC. 

nominator of the fraction. The problem is now reduced to 

A A 

32 _ 32 _ 3 3 £ ;0 _ 1 

F~" ¥~32" T "l6-^ X 7-2 or * 5 ' AnS ' 
16 16 2 

( . 1.25 X 20 X 3 _ In this problem 1.25 X 20 X 

' ' 87+ (11 X 8) " 3 constitutes the numerator of 

459 + 32 the complex fraction. 

1.2 5 Multiplying the factors of the numerator 
X 2 together, we find their product to be 75. 



2 5.0 

75 



87 ! /I 1 y ON 

The fraction — "T^ ' constitutes the denominator of 

the complex fraction. The value of the numerator of this 
fraction equals 87 + 88 = 175. 

The numerator is combined as though it were written 
87 + (11 X 8), and its result is 

11 
8 times 

88 
+ 87 



175 

The value of the denominator of this fraction is equal to 
459 + 32 = 491. The problem then becomes 

3 
75_75 175_75 491 _jT0x 491 _ 1,473 _ 0lft 3 . 
175-T"49T-T X T75- m --y--^ u f Ans - 
491 7 

(75) 1 plus .001 = 1.001. .01 plus .000001 = .010001. 
And 1.001 -.010001 = 

1.001 
.010001 



.990999 Ans. 



ARITHMETIC 

(SECTION 4.) 
(QUESTIONS 76-117.) 



(76) A certain per cent, of a number means so many 
hundredths of that number. 

25fo of 8,428 lb. means 25 hundredths of 8,428 lb. Hence, 
Ufa of 8,428 lb. = .25 X 8,428 lb. = 2,107 lb. Ans. 

(77) Here $100 is the base and Vf> = .01 is the rate. 
Then, .01 X $100 = $1. Ans. 

(78) jr$ means one-half of one per cent. Since l</ is 
2 

.01, \<& is .005, for, 2 )- 010 . And .005 X $35,000 = $175. 
2 - 005 Ans. 

(79) Here 50 is the base, 2 is the percentage, and it is 
required to find the rate. Applying rule, Art. 193, 

rate = percentage -f- base ; 

rate = 2 ~ 50 = .04 or 4#. Ans. 

(80) By Art. 193, rate = percentage -f- base.* 

As percentage = 10 and base = 10, we have rate = 10 
+ 10 = 1 = 100$. Hence, 10 is 100$ of 10. Ans. # 

(81) (#) Rate = percentage -~ by base. Art. 193. 
As percentage = $176.54 and base = $2,522, we have 

rate = 176.54 + 2,522 = .07 = 7$. Ans. 

2 5 2 2 ) 176.54 

.07 



* Remember that an expression of this form means that the first 
term is to be divided by the second term. Thus, as above, it means 
percentage divided by base. 

For notice of the copyright, see page immediately following the title page. 



40 ARITHMETIC. 

(b) Base = percentage -f- rate. Art. 192. 

As percentage = 16.96 and rate = Sfo = .08, we have 

base = 16. 96 -^ .08 = 212. Ans. 
.0 8 ) 1 6. 9 6 
2 12 

(c) Amount is the sum of the base and percentage ; hence, 
the percentage = amount minus the base. 

Amount = 216. 7025 and base = 213. 5 ; hence, percentage = 
216.7025-213.5 = 3.2025. 

Rate = percentage ~- base. Art. 193. 
Therefore, rate = 3.2025 ~ 213.5 = .015 = \\<f>. Ans. 

213.5 ) 3.202 5 ( .015 = 1£0 
2135 



10675 
10675 



(d) The difference is the remainder found by subtracting 
the percentage from the base ; hence, base — the differ- 
ence = the percentage. Base = 207 and difference =201.825, 
hence percentage = 207 - 201.825 = 5.175. 

Rate = percentage -f- base. Art. 193. 

Therefore, rate = 5.175 ~ 207 = .025 = .02^ = fdf. Ans. 

207 ) 5.175 ( .025 
4 14 



1035 
1035 



(82) In this problem $5,500 is the amount, since it 
equals what he paid for the farm -j- what he gained; 
15^ is the rate, and the cost (to be found) is the base. 
Applying rule, Art. 197, 

base = amount -f- (1 -f- rate) ; hence, 

base = $5, 500 -*-■ (1 + . 15) = U, 782. 61. Ans. 



ARITHMETIC. 41 

1,15)5500.0000 ( 4782.61 
46G 



900 
805 



950 
920 



300 
230 



700 
690 



100 
115 



The example can also be solved as follows : 100$ = cost ; if he 
gained 15$, then 100+ 15 = 115$ = $5,500, the selling price. 

If 115$ = $5,500, 1$ = -j- of $5,500 = $47. 8261, and 100$, 

or the cost, = 100 X $47.8261 = $4,782.61. Ans. 



(83) 24 $ of $950 = .24 X 950 = $228 

12-^$ of $950 = .125 X 950= 118.75 
17 $ of $950 = .17 X 950 =161.50 



53-^$ of $950 = $508.25 

The total amount of his yearly expenses, then, is $508.25, 
hence his savings are $950 — $508.25 = $441.75. Ans. 

Or, as above, 24$ + 12 \<f> + 17$ 



53^, 


the 


total 


per- 


-4 


* = 


4*= 


: per 


41.75 = 


- his 


yearly 


sav- 



centage of expenditures; hence, 

cent, saved. And $950 X .465 
ings. Ans. 

(84) The percentage is 961.38, and the rate is-37 g-. By 

Art. 192, 

Base = percentage ^ rate 

= 961.38 -i- .375 = 2,563.68, the number. Ans. 



42 ARITHMETIC. 

Another method of solv- .375)961.38000(2563.6 
ing is the following: 7 5 

If 37 — $ of a number is 2113 



1875 



961. 38, then . 37 - times the 2 3 8 8 

number = 961.38 and the 2 2 50 



number = 961.38 -=- .37^-, *?®° 

2' 1125 

which, as above = 2,563.68. 

Ans. 



2550 
2250 



3000 
3000 

(85) Here $1,125 is 30$ of some number; hence, 
$1,125 = the percentage, 30$ = the rate, and the required 
number is the base. Applying rule, Art. 192, 

Base = percentage -i- rate = $1,125 -^ .30 = $3,750. 

3 1 

Since $3,750 is — of the property, one of the fourths is — 

4 o 

4 
of $3,750 = $1,250, and - or the entire property, is 4 X $1,250 

= $5,000. Ans. 

(86) Here $4,810 is the difference and 35$ the rate. By 
Art. 198, 

Base = difference -s- (1 — rate) 

= $4,810 -T- (1 - .35) = $4,810 4- .65 = $7,400. Ans. 

.65)4810.00(7400 
455 



260 1.00 

260 .35 



00 .65 

Solution can also be effected as follows: 100$ = the sum 
diminished by 35$, then (1 - .35) = .Q5, which is $4,810. 



ARITHMETIC. 43 

If 65 i = $4,810, lfo = -^ of 4,810 = $74, and 100$ = 100 X 

DO 

$74 = $7,400. Ans. 

(87) In this example the sales on Monday amounted to 

$197.55, which was 12- $ of the sales for the entire week; 

Z 

i. e., we have given the percentage, $197.55, and the rate, 

12-$, and the required number (or the amount of sales for 

Z 

the week) equals the base. By Art. 192, 

Base = percentage -f- rate = $197.55 -r- .125; 
or, .125)197.5500(1580.4 Ans. 

125 



725 
625 


1005 
1000 


500 

500 



Therefore, base = $1,580.40, which also equals the sales 
for the week. 

(88) 16.5 miles = 12—$ of the entire length of the road. 

Z 

We wish to find the entire length. 

16.5 miles is the percentage, 12—$ is the rate, and the en- 

Z 

tire length will be the base. By Art. 192, 



Base = percentage ■— rate = 16.5 -^ .12—. 

Z 

.125)16.500(132 miles. Ans. 
12 5 



400 
375 



250 
2 5 J) 

IV.— A 



44 ARITHMETIC. 

(89) Here we have given the difference, or $35, and the 
rate, or 60$, to find the base. We use the rule in Art. 198, 
Base = difference -^ (1 — rate) 

= $35 -T- (1 - .60) = $35 -r- .40 = $87.50. Ans. 
.40)35.000(87.5 
32 



3 00 

2 80 



200 
200 

Or, 100$ = whole debt; 100$ - 60$ = 40$ = $35. 

1 35 

If 40$ = $35, then 1$ = — of 135=—, and 100$ = 

££ X 100 = $87.50. Ans. 
40 

(90) 28 rd. 4 yd. 2 ft. 10 in. to inches. 

X 5£ 

-. ^ , Since there are 5£ yards in 

one rod, in 28 rods there are 
28 X 5i or 154 yards; 154 yards 
15 8 yards plus 4 yards _ 15g yar( } s . There 

are 3 feet in one yard; there- 

4 7 4 fore, in 158 yards there are 

+ 2 3 X 158 or 474 feet; 474 feet + 

2 feet = 476 feet. There are 

12 inches in one foot, and in 

476 feet there are 12 X 476 or 

5712 5,712 inches; 5,712 inches + 10 

~^" 1Q inches = 5,722 inches. Ans. 

5 7 2 2 inches. Ans. 



+ 4 



4 7 6 feet 
X 12 



(91) 12 ) 5 7 2 2 inches. 

3 ) 476 + 10 inches. 
5J ) 158 + 2 feet. 
2 8 + 4 yards. 
Ans. = 28 rd. 4 yd. 2 ft. 10 in. 



ARITHMETIC. 45 

Explanation. — There are 12 inches in 1 foot; hence, in 
5,722 inches there are as many feet as 12 is contained times 
in 5,722 inches, or 476 ft. and 10 inches remaining. Write 
these 10 inches as a remainder. There are 3 feet in 1 yard; 
hence, in 476 feet there are as many yards as 3 is contained 
times in 476 feet, or 158 yards and 2 feet remaining. There 

are 5— yards in one rod; hence, in 158 yards there are 28 rods 
z 

and 4 yards remaining. Then, in 5,722 inches there are 

28 rd. 4 yd. 2 ft. 10 in. 

(92) 5 weeks 3. 5 days. 

X_7 

3 5 days in 5 weeks. 
+ 3.5 

3 8. 5 days. 
Then, we find how many seconds there are in 38. 5 days. 
3 8.5 days 
X 2 4 hours in one day. 

1540 

770 



9 2 4.0 hours in 38.5 days. 
X 6 minutes in one hour. 



5 5 4 4 minutes in 38. 5 days. 
X 6 seconds in one minute. 



3 3 2 6 4 seconds in 38. 5 days. Ans. 

(93) Since there are 24 gr. in 1 pwt., in 13,750 gr. there 
are as many pennyweights as 24 is contained times in 
13,750, or 572 pwt. and 22 gr. remaining. Since there are 
20 pwt. in 1 oz., in 572 pwt. there are as many ounces as 
20 is contained times in 572, or 28 oz. and 12 pwt. remaining. 

Since there are 12 oz. in 1 lb. (Troy), in 28 oz. there are 
as many pounds as 12 is contained times in 28, or 2 lb. and 
4 oz. remaining. We now have the pounds and ounces 
required by the problem; therefore, in 13,750 gr. there are 
2 lb. 4 oz. 12 pwt. 22 gr. 



46 ARITHMETIC. 

24 ) 13750 gr. 

20 ) 5 7 2 pwt. + 22 gr. 
12 )_2_8 oz. + 12 pwt. 
2 lb. + 4 oz. 
Ans. = 2 lb. 4 oz. 12 pwt. 22 gr. 

(94) 100 ) 4763254 li. 

80 ) 47632 + 54 li. 
5 9 5 + 32 ch. 
Ans. = 595 mi. 32 ch. 54 li. 
Explanation. — There are 100 li. in one chain; hence, in 
4,763,254 li. there are as many chains as 100 is contained 
times in 4,763,254 li., or 47,632 ch. and 54 li. remaining. 
Write the 54 li. as a remainder. There are 80 ch. in one 
mile ; hence, in 47, 632 ch. there are as many miles as 80 is con- 
tained times in 47,632 ch., or 595 miles and 32 ch. remaining. 
Then, in 4,763,254 li. there are 595 mi. 32 ch. 54 li. 

(95) 1728 ) 764325 cu. in. 

27 ) 442 + 549 cu. in. 

1 6 cu. yd. + 10 cu. ft. 

Ans. = 16 cu. yd. 10 cu. ft. 549 cu. in. 

Explanation. — There are 1,728 cu. in. in one cubic foot; 
hence, in 764,325 cu. in. there are as many cubic feet as 
1,728 is contained times in 764,325, or 442 cu. ft. and 
549 cu. in. remaining. Write the 549 cu. in. as a remainder. 
There are 27 cu. ft. in one cubic yard ; hence, in 442 cu. ft. 
there are as many cubic yards as 27 is contained times in 
442 cu. ft., or 16 cu. yd. and 10 cu. ft. remaining. Then, in 
764,325 cu. in. there are 16 cu. yd. 10 cu. ft. 549 cu. in. 

(96) We must arrange the different terms in columns, 
taking care to have like denominations in the same column. 

rd. yd. ft. in. 

2 2 2 3 
4 19 

2 7 

3 21 7 

or 3 2 2 1 Ans. 



ARITHMETIC. 47 

Explanation. — We begin to add at the right-hand col- 
umn. 7 + 9 + 3 = 19 in. •, as 12 in. make one foot, 19 in. = 
1 ft. and 7 in. Place the 7 in. in the inches column, and 
reserve the 1 ft. to add to the next column. 

1 (reserved) 4- 2 + 1 + 2 = 6 ft. Since 3 ft. make 1 yard, 
6 ft. = 2 yd. and ft. remaining. Place the cipher in the 
column of feet and reserve the 2 yd. for the next column. 

2 (reserved) + 4 + 2 = 8 yd. Since 5- yd. = 1 rod, 8 yd. = 

1 rd. and 2— yd. Place 2— yd. in the yards column, and 
2 2 

reserve 1 rd. for the next column ; 1 (reserved) -j- 2 = 3 rd. 

Ans. = 3 rd. 2} yd. ft. 7 in. 

or, 3 rd. 2 yd. 1 ft. 13 in. 

or, 3 rd. 2 yd. 2 ft. 1 in. Ans. 

(97) We write the compound numbers so that the units 

of the same denomination shall stand in the same column. 

Beginning to add with the lowest denomination, we find that 

the sum of the gills is 1 -J- 2 -j- 

3 = 6. Since there are 4 gi. in 

1 pint, in 6 gi. there are as many 
pints as 4 is contained times in 
6, or 1 pt. and 2 gi. We place 

2 gi. under the gills column 
and reserve the 1 pt. for the 
pints column; the sum of the 

pints is 1 (reserved) + 5 + 1 + 1 = 8. Since there are 2 pt. 
in 1 quart, in 8 pt. there are as many quarts as 2 is con- 
tained times in 8, or 4 qt. and pt. We place the cipher 
under the column of pints and reserve the 4 for the quarts 
column. The sum of the quarts is 4 (reserved) +8 + 3 = 15. 
Since there are 4 qt. in 1 gallon, in 15 qt. there are as many 
gallons as 4 is contained times in 15, or 3 gal. and 3 qt. re- 
maining. We now place the 3 under the quarts column 
and reserve the 3 gal. for the gallons column. The sum of 
the gallons column is 3 (reserved) + 4+0 + 3 = 16 gal. 
Since we can not reduce 16 gal. to any higher denomination, 
we have 16 gal. 3 qt. pt. and 2 gi. for the answer. 



gal. qt. 


pt. gi. 


3 3 


1 3 


6 


1 2 


4 


1 


8 


5 


16 gal. 3 qt. 


pt. 2 gi. 



48 ARITHMETIC. 

(98) Reduce the grains, pennyweights, and ounces to 
higher denominations. 

24 ) 240 gr. 20 ) 125 pwt. 12 ) 50 oz. 

10 pwt. 6 oz. 5 pwt. 4 lb. 2 oz. 

Then, 3 lb. + 4 lb. 2 oz. -f 6 oz. 5 pwt. + 10 pwt. = 
lb. oz. pwt. 



3 

4 2 



6 5 

10 



deg. 


min. 


sec. 


11 


16 


12 


13 


19 


30 


20 





25 





26 


29 


10 


17 


11 



7 lb. 8 oz. 15 pwt. Ans. 

(99) Since "seconds" is the lowest denomination in 
this problem, we find their sum first, which is 11 + 29 -f- 25 -f 

30 + 12, or 107 seconds. Since 
there are 60 seconds in 1 minute, 
in 107" there are as many minutes 
as 60 is contained times in 107, or 
1 minute and 47 seconds remain- 
ing. We place the 47 under the 
seconds column and reserve the 1 
55° 19' 47" for the minutes column. The sum 

of the minutes is 1 (reserved) -(- 
17 + 26 + 19 + 16, or 79. Since there are 60 minutes in 
1 degree, in 79 minutes there are as many degrees as 60 is 
contained times in 79, or 1 degree and 19 minutes remaining. 
We place the 19 under the minutes column and reserve the 

1 degree for the degrees column. The sum of the degrees 
is 1 (reserved) + 10 + 20 + 13 + 11, or 55 degrees. Since 
we can not reduce 55 degrees to any higher denominations, 
we have 55° 19' 47" for the answer. 

(100) Since "inches" is the lowest denomination in 
this problem, we find their sum first, which is 11 -f 8 + 6, or 
25 inches. Since there are 12 inches in 1 foot, in 25 inches 
there are as many feet as 12 is contained times in 25, or 

2 feet and 1 inch remaining. Place the 1 inch under the 
inches column, and reserve the 2 feet to add to the column 



ARITHMETIC. 49 

of feet. The sum of the feet is 2 feet (reserved) -f- 2 + 1 = 

5 feet. Since there are 3 
rd. yd. ft. in. feet in 1 yard, in 5 feet 

130 5 16 there are as many yards as 3 

215 2 8 is contained times in 5 feet, 

304 4 11 or 1 yard and 2 feet remain- 

650 4- 2 1 i n &- Place the 2 feet under 

mi. ' the column of feet, and re- 

or, 2 10 5 7 Ans. serve the 1 yard to add to the 

column of yards. The sum of 
the yards is 1 yard (reserved) -j- 4 + 5 = 10 yards. Since there 

are 5— yards in 1 rod, in 10 yards there are as many rods as 
Z 

5— is contained times in 10, or 1 rod and 4— yards remaining-. 
Z Z 

Place the 4— yards under the column of yards, and reserve 
Z 

the 1 rod for the column of rods. The sum of the rods is 1 

(reserved) + 304 + 215 + 130 = 650 rods. Place 650 rods 

under the column of rods. Therefore, the sum is 650 rd. 

4— yd. 2 ft. 1 in. Or, since — a yard = 1 ft. 6 in., and since 
Z Z 

there are 320 rods in 1 mile, the sum may be expressed as 

2 mi. 10 rd. 5 yd. ft. 7 in. Ans. 

(lOl) Since "square links" is the lowest denomination 
in this problem, we find their sum first, which is 21 + 23 

+ 16 + 18 + 23 + 21, or 
122 square links. Place 122 
square links under the col- 
umn of square links. The 
sum of the square rods is 
2 + 3 + 2 + 2 + 2 + 3, or 
14 square rods. Place 14 
square rods under the col- 
255 3 14 122 umn of square rods. The 

sum of the square chains 
is 323 square chains. Since there are 10 square chains in 
1 acre, in 323 square chains there are as many acres as 10 is 



A. 
21 


sq. ch. 

67 


sq. rd. 
3 


sq. li. 
21 


28 


78 


2 


23 


47 


6 


2 


18 


56 


59 


2 


16 


25 


38 


3 


23 


46 


75 


2 


21 



50 ARITHMETIC. 

contained times in 323 square chains, or 32 acres and 3 square 
chains remaining. Place 3 square chains under the column 
of square chains, and reserve the 32 acres to add to the col- 
umn of acres. The sum of the acres is 32 acres (reserved) -f 
46 + 25 + 56 + 47 + 28 + 21, or 255 acres. Place 255 
acres under the column of acres. Therefore,, the sum is 
255 A. 3 sq. ch. 14 sq. rd. 122 sq. li. Ans. 

(102) Before we can subtract 300 ft. from 20 rd. 2 
yd. 2 ft. and 9 in., we must reduce the 300 ft. to higher 
denominations. 

Since there are 3 feet in 1 yard, in 300 feet there are 
as many yards as 3 is contained times in 300, or 100 yards. 

There are 5— yards in 1 rod, hence in 100 yards there are as 

Z 

1 11 2 

many rods as 5— or — is contained times in 100 = 18— rods. 

Z Z 11 

100 ■ n -100- 2 _ 100X2 _200 

100. a _100X n - u - 11)20 o(18^rd. 

11 



90 

88 



Since there are 5— or — yards in 1 rod, in — rods there 
Z Z 11 

are -^ X ^f , or one yard, so we find that 300 feet equals 

AA r 
18 rods and 1 yard. The problem now is as follows: From 
20 rd. 2 yd. 2 ft. and 9 in. take 18 rd. and 1 yd. 

We place the smaller number under the larger one, so 
that units of the same denomination fall in the same 
column. Beginning with the lowest 
denomination, we see that inches 
from 9 inches leaves 9 inches. Going 
to the next higher denomination, we 
9 see that feet from 2 feet leaves 
2 feet. Subtracting 1 yard from % 



rd. 


yd. 


ft. 


in. 


20 


2 


2 


9 


18 


1 









ARITHMETIC. 51 

yards, we have 1 yard remaining, and 18 rods from 20 rods 
leaves 2 rods. Therefore, the difference is 2 rd. 1 yd. 2 ft. 
9 in. Ans. 

(103) 



A. 


sq. rd. 


sq. yd. 


114 


80 


25 


75 


70 


30 



39 9 25i Ans. 

Explanation. — Place the subtrahend under the minuend 

so that like denominations are under each other. Then 

begin at the right with the lowest denomination. We 

can not subtract 30 from 25, so we take one square rod 

(= 30j square yards) from 80 square rods, leaving 79 square 
rods ; adding 30-r square yards to 25 square yards, we have 
55-r square yards; subtracting 30 from 55— square yards 

leaves 25— square yards; we now subtract 70 square rods 

from 79 square rods, which leaves 9 square rods; next, we 
subtract 75 acres from 114 acres, which leaves 39 acres, 
which we place under the column of acres. 

(104) If 10 gal. 2 qt. and 1 pt. of molasses are sold 
from a hogshead at one time, and 26 gal. 3 qt. are sold at 
another time, then the total amount of molasses sold equals 
10 gal. 2 qt. 1 pt. plus 26 gal. 3 qt. 

Since the pint is the lowest denomination, we add the 
pints first, which equal + 1, or 1 pint. We can not reduce 

1 pint to any higher denomina- 
tion, so we place it under the 
pint column. The number of 
quarts is 3 + 2, or 5. Since 
1 pt. there are 4 quarts in 1 gallon, 
in 5 quarts there are as many 
gallons as 4 is contained times in 5, or 1 gallon and 
1 quart remaining. We place the 1 quart under the quart 
column, and reserve the 1 gallon to add to the column of 



gal. 


qt. 


pt. 


10 


2 


1 


26 


3 





37 gal. 


1 qt. 


1 



gal. 


qt. 


pt 


62 


3 


2 


37 


i 


1 



52 ARITHMETIC. 

gallons. The number of gallons equals 1 (reserved) -+- 26 
+ 10, or 3? gallons. 

If 37 gal. 1 qt. and 1 pt. are sold from a hogshead of 
molasses (63 gal.), there remains the difference between 63 
gal. and 37 gal. 1 qt. 1 pt. , or 25 gal. 2 qt. and 1 pt. 

63 gal. is the same as 62 gal 3 qt. 2 pt., since 1 gal. equals 

4 qt. and 1 qt. = 2 pt. 

Beginning with the lowest denomination, 1 pt. from the 

2 pt. 1 pint from 2 pints leaves 1 
pint. One quart from 3 quarts 
leaves 2 quarts, and 37 gallons 
from 62 gallons leaves 25 gallons. 

1 Therefore, there are 25 gal. 2 qt. 
and 1 pt. of molasses remaining in the hogshead. Ans. 

(105) If a person were born June 19, 1850, in order to 
find how old he would be on Aug. 3, 1892, subtract the 
earlier date from the later date. 

On August 3, 7 mo. and 3 da. have elapsed from the begin- 
ning of the year, and on June 19, 5 mo. and 19 da. 

Beginning with the lowest denomination, we find that 19 
days can not be taken from 3 days, so we take 1 month from 
7 months. The 1 month which we took equals 30 days, for 

in all cases 30 days are allowed to 
a month. Adding 30 days to the 

3 days, we have 33 days ; subtract- 
ing 19 days from 33 days, we have 

49 i ij. 11 days remaining. Since we bor- 

rowed 1 month from the months 
column, we have 7 — 1, or 6 months remaining; subtracting 

5 months from 6 months, we have 1 month remaining. 1850 
from 1892 leaves 42 years. Therefore, he would be 42 years 
1 month and 14 days old. Ans. 

(106) If a note given Aug. 5, 1890, were paid June 3, 
1892, in order to find the length of time it was due, subtract 
the earlier date from the later date. 

Beginning with the lowest denomination, we find that 5 
can not be subtracted from 3, so we take a unit from the next 



yr. 


mo. 


da 


1892 


7 


3 


1850 


5 


19 



yr. 


mo. 


da. 


1892 


5 


3 


1890 


7 


5 



ARITHMETIC. 53 

higher denomination, which is 
months. The 1 month which we 
take equals 30 days. Adding the 30 
9 28 days to the 3 days, we have 33 days. 

5 days from 33 days leaves 28 days. 
Since we took 1 month from the months column, only 4 
months remain. 7 months cannot be taken from 4 months, 
so we take 1 year from the years column, which equals 12 
months. 12 months -f- 4 months = 16 months. 7 months 
from 16 months = 9 months. Since we took 1 year from the 
years column, we have 1892 — 1, or 1891 remaining. 1890 
from 1891 leaves 1 year. Hence, the note ran 1 year 9 
months and 28 days. Ans. 

(107) Write the number of the year, month, day, hour, 
and minute of the earlier date under the year, month, day, 
hour, and minute of the later date, and subtract. 

22 minutes before 8 o'clock is the same as 38 minutes after 
7 o'clock. 7 o'clock p. m. is 19 hours from the beginning of 
the day, as there are 12 hours in the morning and 7 in the 
afternoon. December is 11 months from the beginning of 
the year. 

10 o'clock a. m. is 10 hours from the beginning of the day. 
July is 6 months from the beginning of the year. The 
minuend would be the later date, or 1,888 years, 11 months, 
11 days, 19 hours, and 38 minutes. 

The subtrahend would be the earlier date, or 1,883 years, 
6 months, 3 days, 10 hours, and 16 minutes. 

Subtracting, we have 

yr. mo. da. hr. min. 
1888 11 11 19 38 
1883 6 3 10 16 



5 5 8 9 22 
or, 5 yr. 5 mo. 8 da. 9 hr. and 22 min. Ans. 

16 minutes subtracted from 38 minutes leaves 22 minutes; 
10 hours from 19 hours leaves 9 hours; 3 days from 11 days 
leaves 8 days; 6 months subtracted from 11 months leaves 
5 months; 1,883 from 1,888 leaves 5 years. 



54 ARITHMETIC. 

(108) In multiplication of denominate numbers, we 
place the multiplier under the lowest denomination of the 
multiplicand, as 

17 ft. 3 in. 

51_ 

879 ft. 9 in. 
and begin at the right to multiply. 51 X 3 = 153 in. As 
there are 12 inches in 1 foot, in 153 in. there are as many 
feet as 12 is contained times in 153, or 12 feet and 9 inches 
remaining. Place the 9 inches under the inches, and reserve 
the 12 feet. 51 X 17 ft. = 867 ft. 867 ft. + 12 ft. (reserved^ 
= 879 ft. 

879 feet can be reduced to higher denominations by divi- 
ding by 3 feet to find the number of yards, and by 5 — yards 
to find the number of rods. 

3 )879 ft. 9 in. 
5.5 )29 3 yd. 

5 3 rd. lj- yd. 
Then, answer = 53 rd. 1| yd. ft. 9 in. ; or 53 rd. 1 yd. 2 
ft. 3 in. 



(109) qt. 

3 


pt. 
1 


gi. 
3 

4.7 


1 8.2 qt. 





.1 


or, 1 8 qt. 


pt. 


1.7 


or, 4 gal. 2 qt. 


pt. 


1.7 



Ans. 

Place the multiplier under the lowest denomination of the 
multiplicand, and proceed to multiply. 4. 7 X 3 gi. = 14.1 gi. 
As 4 gi. =1 pt. , there are as many pints in 14.1 gi. as 4 is 
contained times in 14.1 =3.5 pt. and .1 gi. over. Place .1 
under gills and carry the 3.5 pt. forward. 4.7 X 1 pt. = 4.7 
pt. ; 4.7 + 3.5 pt. = 8.2 pt. As 2 pt. = 1 qt., there are as 
many quarts in 8.2 pt. as 2 is contained times in 8.2 = 4.1 
qt. and no pints over. Place a cipher under the pints, and 
carry the 4.1 qt. to the next product. 4. 7 X 3 qt. = 14.1; 
14.1 + 4.1 = 18.2 qt. The answer now is 18.2 qt. pt. .1 



ARITHMETIC. 55 

gi. Reducing the fractional part of a quart, we have 18 qt. 
pt. 1.7 gi. (.2 qt. = .2 X 8 = 1.6 gi. ; 1.6 + .1 gi. = 1.7 gi.). 
Then, we can reduce 18 qt. to gallons (18 — 4 = 4 gal. and 
2 qt.) = 4 gal. 2 qt. 1.7 gi. Ans. 

The answer may be obtained in another and much easier 
way by reducing all to gills, multiplying by 4.7, and then 
changing back to quarts and pints. Thus, 
3 qt. 
X 2 pt. 3 qt. 1 pt. 3 gi. = 31 gi. 

— q vL 31 gi. X 4.7 = 145.7 gi. 

+ 1 Pt. . 4 )145.7 gi. 

7pt. 2)j36 pt. + 1.7gi. 

X 4 gi. 18 qt. + pt. 

28 Sj- Ans. = 18qt. 1.7 gi. ; 

+ 3 S L or, 4 gal. 2 qt. 1.7 gi. 

31 gi. 
(110) (3 lb. 10 oz. 13 pwt. 12 gr.) X 1.5 = ? 

3 lb. 10 oz. 13 pwt. 12 gr. 
XJL2 

3 6 oz. 
+ 10 

4 6 oz. 
X 20 

9 2 pwt. 
+ 13 

9 3 3 pwt. 
X 24 



2 2 3 9 2 gr. 

+ 12 



2 2 4 4 gr. 

22.404 gr. X 1.5 = 33,606 gr. 

24 ) 3 3 6 6 gr. 
20 )1400 pwt. + 6 gr, 
12 )J_0 oz. + pwt. 
5 lb. + 10 oz. 



56 ARITHMETIC. 

Since there are 24 gr. in 1 pwt., in 33,606 gr. there are as 
many pwt. as 24 is contained times in 33,606, or 1,400 pwt. 
and 6 gr. remaining. This gives us the number of grains 
in the answer. We now reduce 1,400 pwt. to higher denom- 
inations. Since there are 20 pwt. in 1 oz., in 1,400 pwt. 
there are as many ounces as 20 is contained times in 1,400, 
or 70 oz. and pwt. remaining; therefore, there are pwt. 
in the answer. We reduce 70 oz. to higher denominations. 
Since there are 12 oz. in 1 lb., in 70 oz. there are as 
many pounds as 12 is contained times in 70, or 5 lb. 
and 10 oz. remaining. We can not reduce 5 lb. to any 
higher denominations. Therefore, our answer is 5 lb. 10 oz. 
6gr. 

Another but more complicated way of working this 
problem is as follows: 

To get rid of the decimal 
in the pounds, reduce .5 of a 
pound to ounces. Since 
1 lb. = 12 oz., .5 of a pound 
equals .5 lb. X 12 = 6 oz. 
6 oz. -f- 15 oz. = 21 oz. We 
Ans. now have 4 lb. 21 oz. 19. 5 pwt. 
and 18 gr. , but we still have a 
decimal in the column of pwt., so we reduce .5 pwt. to grains 
to get rid of it. Since 1 pwt. =24 gr., .5 pwt. = .5 pwt. 
X 24 = 12 gr. 12 gr. -f 18 gr. = 30 gr. We now have 4 
lb. 21 oz. 19 pwt. and 30 gr. Since there are 24 gr. in 1 pwt. , 
in 30 gr. there is 1 pwt. and 6 gr. remaining. Place 6 gr. 
under the column of grains and add 1 pwt. to the pwt. 
column. Adding 1 pwt., we have 19 + 1 = 20 pwt. Since 
there are 20 pwt. in 1 oz., we have 1 oz. and pwt. remain- 
ing. Write the pwt. under the pwt. column, and reserve 
the 1 oz. to the oz. column. 21 oz. + 1 oz. = 22 oz. Since 
there are 12 oz. in 1 lb., in 22 oz. there is 1 lb. and 10 oz. 
remaining. Write the 10 oz. under the ounce column, and 
reserve the 1 lb. to add to the lb. column. 4 lb. + 1 lb. 
(reserved) = 5 lb. Hence, the answer equals 5 lb. 10 og, 
6gr, 



lb. 


OZ. 


pwt. 


gr. 


3 


10 


13 


1 2 

1.5 


4.5 


15 


19.5 


1 8 


or, 4 


21 


19 


3 


or, 5 


10 





6 



bu. 

2 


pk. 
3 


qt. 
6 
9 


18 


27 


54 


26 


1 


6 



ARITHMETIC. 57 

(111) If each barrel of apples contains 2 bu. 3 pk. and 

6 qt., then 9 bbl. will contain 9 X (2 bu. 3 pk. 6 qt.). 

We write the multiplier under the lowest denomination of 
the multiplicand, which is quarts in this problem. 9 times 
6 qt. equals 54 qt. There are 8 qt. in 1 
pk., and in 54 qt. there are as many pecks 
as 8 is contained times in 54, or 6 pk. and 
6 qt. We write the 6 qt. under the col- 
umn of quarts, and reserve the 6 pk. to 
or 2 6 1 6 a dd to the product of the pecks. 9 times 
3 pk. equals 27 pk. ; 27 pk. plus the 6 pk. 
reserved equals 33 pk. Since there are 4 pk. in 1 bu., in 33 
pk. there are as many bushels as 4 is contained times in 33, 
or 8 bu. and 1 pk. remaining. We write the 1 pk. under 
the column of pecks, and reserve the 8 bu. for the product 
of the bushels. 9 times 2 bu. plus the 8 bu. reserved equals 
26 bu. Therefore, we find that 9 bbl. contain 26 bu. 1 pk. 
6 qt. of apples. Ans. 

(112) (7 T. 15 cwt. 10. 5 lb. ) X 1. 7 = ? When the mul- 
tiplier is a decimal, instead of multiplying the denominate 
numbers as in the case when the multiplier is a whole num- 
ber, it is much easier to reduce the denominate numbers to 
the lowest denomination given ; then, multiply that result 
by the decimal, and, lastly, reduce the product to higher- 
denominations. Although the correct answer can be ob- 
tained by working examples involving decimals in the 
manner as in the last example, it is much more complicated 
than this method. 7 T. 15 cwt. 10.5 lb. 

X _20 

14 cwt. 
15 



15 5 cwt. 

X 100 



15 5 lb. 

10.5 
15510.5 lb. 
15,510.5 lb. X 1.7 = 26,367.85 It?, 



58 ARITHMETIC. 

There are 100 lb. in 1 cwt., and in 26,367.85 lb. there are 
as many cwt. as 100 is contained times in 26,367.85, which 

equals 263 cwt. and 67.85 lb. 

100 ) 2 6 3 6 7.8 5 lb. remaining. Since we have 

20 )263 cwt74- 67.85 lb. the number of pounds for 

1 3 T. -f 3 cwt. our answer, we reduce 263 

cwt. to higher denominations 

There are 20 cwt. in 1 ton, and in 263 cwt. there are as 

many tons as 20 is contained times in 263, or 13 tons and 3 

cwt. remaining. Since we cannot reduce 13 tons any 

higher, our answer is 13 T. 3 cwt. 67.85 lb. Or, since .85 

lb. = .85 lb. x 16 = 13.6 oz., the answer may be written 

13 T. 3 cwt. 67 lb. 13.6 oz. 

(113) 7 ) 358 A. 57 sq. rd. 6 sq. yd. 2 sq. ft. 

51 A. 31 sq. rd. sq. yd. 8 sq. ft. Ans. 

We begin with the highest denomination, and divide each 
term in succession by 7. 

7 is contained in 358 A. 51 times and 1 A. remaining. 
We write the 51 A. under the 358 A. and reduce the remain- 
ing 1 A. to square rods = 160 sq. rd. ; 160 sq. rd. + the 57 
sq. rd. in the dividend = 217 sq. rd. 7 is contained in 217 
sq. rd. 31 times and sq. rd. remaining. 7 is not contained 
in 6 sq. yd., so we write under the sq. yd. and reduce 
6 sq. yd. to square feet. 9 sq. ft. X6= 54 sq. ft. 54 sq. ft. 
+ 2 sq. ft. in the dividend = 56 sq. ft. 7 is contained in 
56 sq. ft. 8 times. We write 8 under the 2 sq. ft. in the 
dividend. 

(114) 12 ) 282 bu. 3 pk. 1 qt. 1 pt. 

23 bu. 2 pk. 2 qt. J pt. Ans. 
12 is contained in 282 bu. 23 times and 6 bu. remaining. 
We write 23 bu. under the 282 bu. in the dividend, and 
reduce the remaining 6 bu. to pecks — 24 pk. -f- the 3 pk. in 
the dividend = 27 pk. 12 is contained in 27 pk. 2 times and 
3 pk. remaining. We write 2 pk. under the 3 pk. in the 
dividend, and reduce the remaining 3 pk. to quarts. 3 pk. 
= 24 qt. ; 24 qt. + the 1 qt. in the dividend = 25 qt. 12 is 
contained in 25 qt. 2 times and 1 qt. remaining. We write 



ARITHMETIC. 59 

2 qt. under the 1 qt. in the dividend, and reduce 1 qt. to 
pints = 2 pt. + the 1 Pt- i n the dividend = 3 pt. 3 -f- 12 = 

3 1 4. 

12 ° r 4 Pt 

(115) We must first reduce 23 miles to feet before we 
can divide by 30 feet. 1 mi. contains 5,280 ft. ; hence, 23 
mi. contain 5,280 X 23 = 121,440 ft. 

121,440 ft."-*- 30 ft. = 4,048 rails for 1 side of the track. 
The number of rails for 2 sides of the track = 2 X 4,048, 
or 8,096 rails. Ans. 

(116) In this case where both dividend and divisor are 
compound, reduce each to the lowest denomination men- 
tioned in either and then divide as in simple numbers. 

1 bu. 1 pk. 7 qt. 3 5 6 bu. 3 pk. 5 qt. 

X 4 X 4 



4 pk. 14 2 4 pk. 
+ 1 +3 

5 pk. 14 2 7 pk. 
X 8 X 8 

1 1 4 1 6 qt. 

+ 5 





4 qt. 






+ ? 






47 qt. 




) 


11421( 
94 

202 


243 




188 


- 




141 






141 





11421 qt. 



11,421 qt.-*- 47 qt. = 243 boxes. 

Ans. 



(117) We must first reduce 16 square miles to acres. 
In 1 sq. mi. there are 640 A., and in 16 sq. mi. there are 
16 X 640 A. = 10,240 A. 

62 ) 1 2 4 A. 

165 A. 25sq.rd. 24sq.yd. 3 sq.ft. 80+sq. in. Ans 
Q. G. IV.— 5 



60 ARITHMETIC. 

62 is contained in 10,240 A. 165 times and 10 A. remain- 
ing. We write 165 A. under the 10,240 A. in the dividend 
and reduce 10 A. to sq. rd. In 1 A. there are 160 sq. rd., 
and in 10 A. there are 10 X 160 = 1,600 sq. rd. 62 is con- 
tained in 1,600 sq. rd. 25 times and 50 sq. rd. remaining. 
We write 25 sq. rd. in the quotient and reduce 50 sq. rd. to 

sq. yd. In 1 sq. rd. there are 30 — sq. yd., and in 50 sq. rd. 
there are 50 times 30 — sq. yd. — 1,512— sq. yd. 62 is con- 
tained in 1,512— sq. yd. 24 times and 24— sq. yd. remaining. 

2 2 

In 1 sq. yd. there are 9 sq. ft., and in 24— sq. yd. there are 

2 

24- X 9 = 220- sq. ft. 62 is contained in 220- sq.ft. 3 times 

2 2 2 

and 34- sq. ft. remaining. We write 3 sq. ft. in the quo- 
2 

tient and reduce 34- sq. ft. to sq. in. In 1 sq. ft. there are 
2 

144 sq. in., and in 34- sq. ft. there are 34- X 144 = 4,968 
A 2 

sq. in. 62 is contained in 4,968 sq. in. 80 times and 8 sq. in. 

remaining. 

We write 80 sq. in. in the quotient. 

It should be borne in mind that it is only for the purpose 
of illustrating the method that this problem is carried out 
to square inches. It is not customary to reduce any lower 
than square rods in calculating the area of a farm. 



ARITHMETIC. 

(SECTION 5.) 
(QUESTIONS 118-133.) 



(118) To square a number, we must multiply the num. 
ber by itself once, that is, use the number twice as a factor 
Thus, the second power of 108 is 108 X 108 = 11,664. Ans 



108 
108 

864 
1080 

11664 



(119) 9 5 = 9 X 9 X 9 X 9 X 9 = 59,049. Ans. 



729 
9 

6561 
9 

59049 

(120) (a) .0133 3 = . 0133 X. 0133 X. 0133 = . 00000235263? 

Ans. 

For notice of copyright, see page immediately following the title page. 



62 ARITHMETIC. 

Since there are four decimal places in the multiplicand and 
4 in the multiplier, we must point off 4 + 4=8 decimal 

places in the product; but as 

.0133 there are only 5 figures in the 

.0133 product, we prefix three ciphers 

399 to form the eight necessary deci- 

399 mal places in the first product. 

X 3 3 Since there are 8 decimal places 

Q in the multiplicand and 4 in the 

multiplier, we must point off 8 

! 1 +4 = 12 decimal places in the 

5 3067 product; but as there are only 

5 3 6 7 7 figures in the product, we prefix 

17 6 8 9 5 ciphers to make the 12 neces- 



.00000235 2 637 sai T decimal places in the final 
product. 

(121) Evolution is the reverse of involution. In invo- 
lution we find the power of a number by multiplying the 
number by itself one or more times, while in evolution we 
find the number or root which was multiplied by itself one or 
more times to make the power. 

(122) 4/90 = ? The root is evidently 9 plus an intermi- 
nable decimal. Trying 9 for one factor, the other is 90 -=- 9 

= 10, and the first approximation is- — = 9.5. 90 -f- 9.5 

2 

= 9.473+, and the second approximation is — — ~±— - = 

2 

9.486+, or 9.49 to three figures. Using 9.49 for one fac- 
tor, the other is 90^9.49 = 9.48366 + , and the third 

. 9.49 + 9.48366 

approximation is -+- =9.48683 or 9.4868+ to 

2 

five figures. Ans. 

This solution may be shortened by using the table and 

applying the method described in Art. 281 to find the first 

three significant figures of the root. Referring to the table, 

the first two significant figures of the root are 9.4; the first 

difference is 90.25 — 88.36 = 1.89; the second difference is 



ARITHMETIC. 63 

90-88.36 = 1.64; 1.64 -4- 1.89 = , 86+. Hence, the first 
three figures are 9.48. 

( 1 23) To find any power of a mixed number, first reduce 
it to an improper fraction, and then multiply the numerators 
together for the numerator of the answer, and multiply the 
denominators together for the denominator of the answer 



('!)' " T 



5 15 15 _ 15 X 15 X 15 3.375 _ 47 
X 4 X 4 : 4X4X4 64 = 64 



= 52.734375. Ans. 



3 = 3X4 + 3 = 12 + 3 15 

47 
15 64)3375(5 2~ 

15 3 2 



75 175 

15 128 



225 47 

15 



1125 
225 

3375 



4) 47.000000 ( .734375 
448 

"Tio 

192 

Tso 

256 

Tio 

192 

480 

448 

~2 

32 



64 ARITHMETIC. 

Since six ciphers were annexed to the dividend, six decimal 
places must be pointed off in. the quotient. 



(1 24) y 92,416 = ? Pointing off into periods, the result 

is 92'416. Since 92 lies between 4 3 = G4 and 5 3 = 125, the 

root is 4-f-. Trying 4 for one cf the two equal factors, the 

third factor is 92 -r- 4" = 92 -=- 16 = 5.75. Trying 5 for one 

of the two equal factors, the third factor is 92 -=- 5 2 = 92 -f- 25 

= 3.68. Difference between 4 and 5.75 is 1.75, and between 

5 and 3.68 is 1.32; hence, use 5, the first approximation 

,. 2 X 5 + 3.68 . „. . a . + fl 

being ^ = 4.56, or 4.6 to two figures. 

o 

Using 46 for one of the two equal factors, the third factor 

is 92416 -f- 46 2 = 92416 -r- 2116 = 43.67+, and the second 

. . . 2X 46 + 43.67 .. K 00 . AK _' . 

approximation is ^ = 45.22+, or 45.2 to three 

o 

figures. 

Using 45.2 for one of the two equal factors, the third 

factor is 92416 -h 45. 2 2 = 92416 -r- 2043.04 = 45.2345 + , and 

.u .u- a ■ ♦■ -2x45.2 + 45.2345 + 

the third approximation is ■ = 4o.2115 + , 

o 

or 45.212— to five figures. Ans. 

The first three significant figures may also be found by the 
aid of the table and the method described in Art. 282. 

In order to obtain two figures of the root from the table, 
we place a decimal point between the first and second signifi- 
cant periods ; the result is 92.416. Referring to the table, the 
first two figures of the root are 4.5; the first difference is 
97.336-91.125 = 6.211; the second difference is 92.416 
- 91.125 = 1. 291; 1 .291 -h 6.211 =.20+. Therefore, ^92.416 
= 4.52 and V^2,416 = 45.2 to three significant figures. 

(1 25) 4/502,681 = ? Pointing off into periods, we have 
50'26'81. The first figure of the root is evidently 7, since V 
= 49 and 8 a = 64. The two factors then are 7, and 50 -f- 7 

= 7. 14+. The first approximation is 7 + 7 ; 14+ = 7.07+, 

Z 

or 7.1 to two figures. To find the second approximation, we 
use the first two periods and drop the decimal point in the first 






ARITHMETIC. 65 

approximation. One factor is then 71 and the other 5026 
-f- 71 = 70.7S + . The second approximation is therefore 
71 +^70.78 _ 70>89 ^ or 70>9 to three fig Ures . Using 709 for 

one factor, the other is 502681 -+ 709 = 709. Hence, the 
number is a perfect power and the root is 709. Ans. 

Using the table and placing a decimal point between the 
first and second periods so that we may obtain two figures of 
the root from the table, the number becomes 4/50.268I or to 
four figures, j/50.27. Referring to the table, the first two 
figures of the root are 7. ; the first difference is 50. 41 — 49. 00 
= 1.41; the second differenc e is 50.2 7 - 49.00 = 1.27; 1.27 
~- 1.41 = .9-1-. Therefore, 4/502,681 = 709 to three figures. 

(126) 3/!*p 3 Ans< 

(127) 4 3 = 4 X 4 X 4 = 64. 

4^8 = 2. 
4 3 - 4/8 = 64 - 2 = 62. Ans. 

(128) Since? =.375,^1 = ^^75. Moving the deci- 

mal point three places to the right, the number becomes 375. 
Since 375 lies between V = 343 and 8 3 = 512, the root is 7+. 
Trying 7 for one of the two equal factors, the third factor is 
375 -r- V — 375 -T- 49 = 7.65+, and the first approximation is 

2X7 + 7.6o _ 7>21+> or 7>2 to two figures> As tll e differ- 

o 
ence between the equal and unequal factors is very slight, it 
is not necessary to try 8. 

Using 7.2 for one of the two equal factors, the third 
factor is 375 -r- 7.2 2 = 375 -f- 51.84 = 7.233+, and the second 

. /0 X 7./C + 7.2oo w w . 

approximation is — = 7.211+, or 7.21 to three 

o 

figures. 

Using 7.21 for one of the two equal factors, the third 

factor is 375 -5- 7.21* = 375 -j- 51.9841 = 7.21374+, and the 

f , . A . . . 2X 7.21 + 7.21374 - 0110 - . 
third approximation is = 7.21124+, or 



G6 ARITHMETIC. 

7.2112+ to five figures. Since the number is entirely deci 
mal, the root is wholly decimal; hence, locating the decimal 
point, the \/lm = .72112+. Ans. 

By aid of the table, the first three figures are determined 
as follows: Move the decimal point three places to the right 
so it will fall between the first period 375 and the cipher 
period that follows. 

Referring to the table, the first two figures of the root are 
7.2; the first difference is 389. 17 — ^7 \248 = 15.769; the 
second difference is 375.000 - 373.248 = 1.752 • 1.752 
-s- 15.769 = .11 + , or .1 to one figure. Hence, the first three 
figures are 7.21, or the J/J575 = .721. The fourth and fifth 
figures are then determined as previously indicated. 



(129) 4/. 3364 = ? 

Moving the decimal point two places to the right, so that 
the first period may be integral, the result is 33. 64. The first 
two factors are evidently 5 and 33 ■ -f- 5 = 6. 6, and the first 

K 1 C (I 

approximation is — — = 5.8. 33. 64 -f- 5. 8 = 5. 8. Hence, 

the given number is a perfect power, and as it is wholly 
decimal, 4/. 3364 = .58. Ans. 



(130) */3.1416 = ? 

Pointing off, we obtain 3.14'16. The first two significant 

figures are 3.1. It is evident that the first figure of the root 

is 1, since l 2 = 1 and 2 2 = 4. Using 1 as one factor, the other 

1+ 3 I 
is 3.1 -r- 1 = 3.1, and the first approximation is = 2.05. 

2 

Had 2 been used as one factor, the other would have been 

3.1 -f- 2 = 1.55, and the first approximati n would have been 

2 + 1 55 

' = 1.77+, or 1.8 to two figures. In the first case, 

Z 

the difference between the two factors is 3.1 — 1 = 2.1; in 
the second case, the difference is 2 —.1.55 = .45. As the 
factors are more nearly equal in the second case than in the 
first, it is evident that 1.8 is more nearly equal to the correct 
value of the root than 2.05 is; hence, 1.8 will be used for the 
first approximation. 



ARITHMETIC. 67 

For the second approximation, use the first two periods 

and 1.8 for one factor, the other factor is 3. 1 4 .-£- 1.8 — 1. 74+ ; 

18 4-1 74 
hence, the second approximation = — — : — =1.77. Using 

2 

1.77 for one factor, the other is 3.1416 -f- 1.77 = 1.77491 +. 

1.77 + 1 77491 
The third approximation is — - 1 = 1.772455, or 

2 

1.7725— to five figures. Ans. 

Using the table to find the first three figures, the first two 
figures of the root are 1.7; the first difference is 3.24 — 2.89 
= .35; the second difference is 3.14 — 2.89 = .25 ; .25 -^ .35 
= .71+. Therefore, ^3.1416 = 1.77 to three significant 
figures. 

(131) Since some number multiplied by itself equals 

114.9184, then the number is |/ll4.9184. Pointing off into 
periods and placing the decimal point between the first and 
second periods, we have 1.14'91'84. Considering the first 
two figures, it is evident that the first figure of the root is 1. 
Using 1 as one factor, the other is 1.1 -r-'l = 1.1, and the first 

approximation is — — — = 1.05, or 1.1 to two figures. Using 

2 

the first three figures and 1.1 for one factor, the other factor 
is 1.15 -f- 1.1 = 1.045+, and the second approximation is 

1.1 + 1.045 = lo72 _j_ j or L07 t0 three fi gures> Using 1.07 

2 

for one factor, the other is 1.149184 -h 1.07 = 1.074003 + , 

.. lt !. . . . 1.07 + 1.074003 , AWn/wvl , 
and the third approximation is = 1.072001+, 

2 

or 1.0720 to five figures. Noticing that the square of the last 
significant figure is 2 2 = 4, which corresponds to the last figure 
of the given number, and that the fifth and sixth figures of the 
third approximation are ciphers, we suspect that the given 
number is a perfect power. We find such to be the case on 
squaring 1.072. Since there are two periods in the integral 
part of the number, there are two figures in the integral part 
of the root, and 4/114.9184 = 10.72. Ans. 

Using the table to find the first three figures of the root, 
the first two figures of the root are 1.0; the first difference 



08 ARITHMETIC. 

is 1.21 — 1.00 = .21; the second difference is 1.15—100 
= ,15; .15 -T- .21 = .7. Therefore, the first three figures 
are 10.7. 



(132) 4/3,486,784 = ? 

Pointing off into periods, we have 3'48'67'84. Placing a 

decimal point between the first and second periods and 

using the first two figures, we obtain 3.5. Considering 2 as 

one factor, the other is 3.5 -f- 2 = 1.75, and the first approxi- 

2 4-1 75 

mation is - 1 — = 1.87+, or 1.9 to two figures. For the 

2 

second approximation, we use the first two periods and 19 for 

one factor, the other factor being 349 ~ 19 = 18.37 + . We 

used 349 instead of 348 because the fourth figure was 6, and 

the number correct to three figures is 349. The second 

approximation is — : — = 18.68+, or 18,7 to three 

figures. 34868-^187 = 186.459+; 187 + 186 - 459 =186.729+, 

2 

or 1867.3— to five figures. Ans. 

In order to obtain three figures from the table, we place 
the decimal point between the first and second periods, 
and use the first two periods only ; that is, we find the value 
of 4/3.49. Referring to the table, the first two figures of the 
root are 1.8; the first difference is 3.61 — 3.24 = .37; the 
second difference is 3.49 - 3.24 = .25; .25 ~ .37 = .67+, 
or .7 to one figure. Therefore, 4/3,486,784 = 187 to three 
significant figures. 

The fourth and fifth figures may be found as previously 
indicated. 



(133) 4/. 00041209 = ? 

Pointing off into periods, the result is .00'04'12'09. Placing 
the decimal point between the first and second periods of the 
significant part of the number, we obtain 4.1 for the first two 
figures. The first factor is evidently 2 and the second factor 

O_|_0 ()K 

4. 1 -=- 2 = 2. 05. The first approximation is "*" ~" = 2. 02+ , 

2 

or 2.0 to two figures. Using 20 for one factor, the other is 



ARITHMETIC. 69 

412 -f- 20 = 20.6, and the second approximation is — -^- — — 

= 20.3. 

Using 203 for one factor, the other is 41209 -r- 203 = 203. 
Hence the number is a perfect power and the significant 
figures of the root are 203. There being one full cipher period 
following the decimal point, the root is .0203. Ans. 

Using the table, we find that the first two figures of the 
root are 2.0; the first difference is 4.41 — 4.00 =.41 ; the sec- 
ond difference is 4.12 — 4.00 = .12; .12 -h .41 = .3, nearly. 
Therefore, 4/. 00041209 = .0203 to three significant figures, 



ARITHMETIC 

(SECTION 6.) 
(QUESTIONS 143-167.) 



(143) 11.7 : 13:: 20 : x. The product of the means 

11.7*= 13x20 equals the product of the 

11.7*= 260 extremes. 

_ 260 
*~1L7) 260.000 (22.22+ Ans. 
234 







260 






234 




260 






234 




260 






234 






~26 


(144) 


(a) 20 + 7 : 10 + 8:: 3: x. 




27 : 


18:: 3 : x 




27* 


= 18 X 3 




27* 


= 54 




X 


= g = 2. Ans. 




(b) 12 a : 


100 2 ::4 : x. 




144 : 


10,000:: 4 : x 




144 * = 10,000 X 4 




144*= 40,000 


For noti 


ce of copyri 


ght, see page immediately following the title page. 



x = 



ARITHMETIC. 
40,000 



144 ) 40 0.0 ( 277.7+ Ans. 
288 
1120 
1008 



1120 
1008 



1120 

1008 

112 

4 7 
(145) (a) - = — , is equivalent to 4 : *::? : tfl. The 

product of the means equals the product of the extremes. 
Hence, 

7x- 4X 21 

7 ;r = 84 

84 
x — — or 12. Ans. 

(£) In like manner, 

x 8 

^r-: = — is equivalent to x : 24 :: 8 : 16 
*4: lb 

16 x= 24 X 8 
16* = 192 

.r = — - = 12. Ans. 
lb 

2 x 

(c) — = — - is equivalent to 2 : 10 :: x : 100. 

10* = 2 X 100 
10* = 200 

* = ^ = » Ans. 

, ,* 15 60. . . , x 10 x . . , 

(« ) -= = — is equivalent to (e) -rx = 777777 is equivalent to 
' 4o * n w loO 600 

15 : 45 :: 60 : x. 10 : 150 :: x : 600. 

15 * = 45 X 60 150 x = 10 X 600 

15 * = 2, 700 150.* = 6,000 

2 i700 1cm 6,000 tf% A 

*=-i-— =180. ^r = -i— -^.40. Ans. 
15 . 150 

Ans. 



ARITHMETIC. 73 



(146) x\ 5 :: 27 : 12.5. (147) 


45 : GO :: x : 24 


5 


60 x = 45 X 24 


12.5)135.0(10- Ans. 
125 5 


60 * = 1,080 

^'f^ 80 =18. Ans. 


100_ 4 


60 


lT5 _ 5 




(148) x\ 35 :: 1 : 7. (149) 


9 : x :: 6 : 24. 


7 * = 35 X 4 


6j=9x24 


7.r = 140 


6* = 216 


* = ~ = 2(T Ans. 


* = ^ = 36. Ans, 




(150) 




^1,000 : f 1,331 :: 27 : x. 


^1,000 = 10. 


10 : 11:: 27 : x. 


^1,331 = 11. 


10,r =297. 1 


1 1'331(11 


*=* 9 n 7 =29.7. I 


2 1 


10 9 

Ans. * 


3 3 3 1 


31 331 


3 


331 


1 





31 
(151) 64 : 81 = 21 2 : x\ 

Extracting the square root of each term of any proportion 
does not change its value, so we find that |/64 : 4/8I = 
4/2I 2 : \/~~x' 1 is the same as 
8 : 9 



(152) 7 + 



> = 21 : x 






Sx= 189 






„r=23.625. Ans. 




: 7 = 30 : x 


is equivalent 


to 


15 : 7: 


= 30 : x. 




15 x = 


7 X 30 




15 x = 


210 




x = 


= ^5- =14. 

15 


Ar 



74 ARITHMETIC. 

(153) 2 ft. 5 in. = 29 in. ; 2 ft. 7 in. = 31 in. Stating 
as a direct proportion, 29 : 31 = 2,480 : x. Now, it is easy 
to see that x will be greater than 2,480. But x should be 
less than 2,480, since, when a man lengthens his steps, the 
number of steps required for the same distance is less; 
hence, the proportion is an inverse one, and 

29 : 31 = x : 2,480, 
or, 31^ = 71,920; 

whence, x = 71,920 ~- 31 = 2,320 steps. Ans. 

(154) This is evidently a direct proportion. 1 hl\ 
36 min. = 96 min. ; 15 hr. = 900 min. Hence, 

96 : 900 = 12: x, 
or, 96 x = 10,800; 

whence, x = 10,800 -=- 96 = 112.5 mi. Ans. 

(155) This is also a direct proportion ; hence, 

27.63 : 29.4 = .76 : x, 
or, 27. 63 x = 29. 4 X . 76 = 22. 344 ; 

whence, x = 22. 344 ~ 27. 63 = . 808 + lb. Ans. 

(156) 2 gal. 3 qt. 1 pt. = 23 pt. ; 5 gal. 3 qt. = 46 pt. 

Hence, 

23 : 46 = 5 : x, 

or, 23;r= 46 X 5 = 230; 

whence, x = 230 -~ 23 = 10 days. Ans. 

(157) Stating as a direct proportion, and squaring the 
distances, as directed by the statement of the example, 
6" : 12 2 = 24 : x. Inverting the second couplet, since this 
is an inverse proportion, 

6' : 12* = x : 24. 

Dividing both terms of the first couplet (see Art. 310) 
by 6 

l a : r = x : 24; or 1 : 4 = x : 24; 
whence, 4 x = 24, or x = 6 degrees. Ans, 



ARITHMETIC. 



75 



(158) Taking the dimensions as the causes, 



n 


15 




2 




x, whence, 2 x = 75, or, x = $37. 50. 

Ans. 



or 



(159) 2 hr. =120 min. ; 14 hr. 28 min. = 868 min. 
Hence, 120 : 868 = 100 : x, 

or, 120.2- = 86,800; 

whence, x = 723^ gal. Ans. 

^160) Taking the dimensions as the causes, 



x, whence, 2 x = 17 X 57 = 969, 
or, x = 484^ bbl. Ans 



(161) 8 hr. 40 min. = 520 min. Hence, 

444 : 1,060 = 520 : x, 
130 

, x = 1.Q6QX m = 15M00 m 1)24L44 + m . n> = 20 hr< 

ffi U1 41.44+ min. Ans. 

(162) 1 min. = 60 sec. Hence, 

5} : 60 = 6,160 : *, 
60 X 6,160 



w 


* 


# 


# 


2 


=w 


;2 


17 57 


» 






or, .ar 



5.5 



67,200 ft. Ans. 



(163) Writing the statement as a direct proportion, 
8 : 10 = 5 : x, it is easy to see that x will be greater than 5; 
but, it should be smaller, since by working longer hours, 
fewer men will be required to do the same work. Hence, 
the proportion is inverse. Inverting the second couplet, 

8:10 = ^: 5, 
4 



or, 



x = 



.9. 



= 4 men. Ans. 



a. a. iv.— 6' 



76 



ARITHMETIC. 



(164) Taking the times as the causes, 



99 


& 


14 







n 




=m 


m 


19 


x P 




2 


3 





ffip ; whence, 3x — 2 X 14 = 28, or x = 9£ hr. 

Ans. 



(165) Taking the horsepowers as the effects, we have 
for the known causes in example 4, Art. 349, 14 2 , 500, and 
48, and for the known effect 112 horsepower. Hence, 









9 




w 


m 




m 


22 


14 3 


30 2 


t 


lit 


500 


660 = 112 


x, or fflty 


M = ll9 


48 


42 


$ 


3 £ 






» 


n 



whence, j=9x22X3 = 594 horsepower. Ans. 

(166) First find the volume of the cylinder in cubic 
inches, as in the example, Art. 345. The volume, multi- 
plied by the weight of one cubic inch (.261 lb. ), will evidently 
be the weight of the cylinder. Hence, 



10 2 


12 2 


20 


60 


hence, . 


x = 



1,570.8 



x, or 



100 



99 



144 



= 1.570.8 



x\ 



144 X 3 X 1,570.8 




6,785.856 cu. in. Therefore. 



weight of cylinder = 6,785.856 X .261 = 1,771.11 lb. Ans 
(167) Referring to the example in Art. 348, 

5 n 
w 

100 
x, or $>p 324 = 187 

19 



324 
4 

19 



u 324 X 4 x 187 
whence, x = - - = 484. 7 lb. Ans. 



NOTICE. 

The present set of answers on the subject of Arithmetic are 
less in number than were contained in the former edition. 
As a consequence, there is a slight break in the page numbers 
between the last page of the answers on Arithmetic and the 
paper following. 






ALGEBRA. 

(QUESTIONS 168-217.) 



(169) (a) Factoring each expression (Art. 457), we 
have 9x* + 12^y+ 4/ = (3.r 2 + 2/) (3x 2 + 27 2 ) = (3;tr 2 + 2/)\ 

(b) 49tf 4 - 154tf 2 £ 2 + 12l£ 4 = (la" - lib") (la" - lib") = 
(la" - lib")". Ans. 

(c) Ux"y" + Uxy + 16 = 16(2^ + l) 2 . Ans. 

(170) (#) Arrange the dividend according to the 
decreasing powers of x and divide. Thus, 

3x - 1 ) 9x 3 + 3x" + x - 1 ( 3-r* + 2x + 1 Ans. 



(Art. 444.) 





9;r 3 


- 3x" 






6x" + X 






6x* — 2x K 






Sx-1 






3x-l 


tf 


-') 


a 2 - 2ab" + b* ( a" + 
a* - a"b 




a"b - 2ab" 






a"b- ab" 




- ab"-\-b 3 






- ab" + b* 



(b) a — b)a s - 2ab" + b' 6 (a" + ab - b" Ans. 

(Art. 444.) 



(c) Arranging the terms of the dividend according to 
the decreasing powers of x, we have 

Ix - 3 ) lx* - 24;r 2 + 58;r - 21 ( x* - 3x -f 7 Ans. 
lx 3 - 3.r 2 

- 21^ 2 + 58;r 
-21^ 2 + 9;r 



49.r - 21 
49£--21 

For notice of the copyright, see page immediately following the title page. 



06 ALGEBRA. 

(171) See Arts. 352 and 353. 

(172) (a) In the expression 4x 3 y — V&sfy 1 -f- 8xy", it is 
evident that each term contains the common factor kxy. 
Dividing the expression by 4-ry, we obtain x 2 — 3x 2 y -(- 2y* 
for a quotient. The two factors, therefore, are Axy and 
x* - 3x*y + 2y\ Hence, by Art. 452, 

±x 3 y - 12x 3 y* + 8^r 3 = ±xy(x* - 3x'y + 2/). Ans. 
(£) The expression (x* — y*) when factored, equals (x % -f- 
/)(*' - /)• ( Art - 463.) But, according to Art. 463, x* 
—y 1 may be further resolved into the factors (x -\-y)(x — y). 

Hence, (x 4 - y*) = (x* +y*)(x -\-y)(x - y). Ans. 

(c) 8.r 3 — 27/. See Art. 466. The cube root of the 
first term is 2x, and of the second term is 3y, the sign of the 
second term being — . Hence, the first factor of 8x 3 — 21y 3 
is 2x — 3y. The second factor we find to be 4;r 2 -f- 6xy + 9j 5 , 
by division. Hence, the factors are 2x — 3y and 
4;r 9 + 6xy + 9y\ 

(173) Arranging the terms according to the decreasing 
powers of m. 

3m 3 + 10m 2 72 + 10/nn* + 3n % 
3m 4 n — 5//i 3 7i 2 -\- 5//t 2 7i 3 — mn* 

9m 7 7i + 30wV + 30//zV + 9/#V 

— 15/«V - 5077l b 7Z 3 — 507/1*71* — 15?«V 

+ 157/zV + 507/1*71* + 50»zV + 15//rV 

- 377i*7i* — 10;;* V - 10;;?V — 37/in 1 

$m 7 n -f- 15;;z 6 ;z 2 — 5m*n* -f- 6*# 4 » 4 + 257/i 3 n r ° + 5//z 2 # 6 — 3w;z 7 

Ans. 

(174) (20'&r 8 ) 4 = lGtf 8 £V 12 . Ans. 

( - 3a 2 b 2 c) b = - 2±3a 10 b i0 c\ Ans. 
( — 77n 3 nxy* y = 4:97n 6 n 7 x*y\ Ans. 

(175) (a) 4a* ~ ^factored = {2a + b)(2a - b). Ans. 

(b) 16x 10 - 1 factored = (4;tr 5 + l)(4;r 5 - 1). (Art. 463.) 

Ans. 



ALGEBRA. 97 

(c) 16x 9 — 8^y + x*y\ when factored = 

(4.r 3 - xf)(4:X 3 - xy*). (Art. 457, Rule.) 

But, (4.r 3 — xy 2 ) = x(2x + y)(2x — y). (Arts. 452 and 
463.) 

Hence, 16^- 6 - 8^> 3 + *V = x* (2x +y) (2x + j) (2* — j) 
(2x—y). Ans. 

(1 76) 4tf 6 — 12tf 5 .r -J-5« 4 ;r a +6tf 3 ^ 3 +tfV 4 (2tf 3 — 3# 2 ;tr— «;tr 3 
4# 6 Ans. 



4tf 3 — 3tf 2 .r 



4tf — 6a x— ax' 



— 12« 5 ^r + 5aV 

- 12tf 5 ;tr + 9tfV 



4<2 4 .r 2 -f- 6# 3 ;i' 3 4- a 2 x i 



(177) (*) 6# 4 3 4 + ? 3 £ 3 — la'P-^ZaAc+S, 

(b) 3 + 2« ^ + a 5 b 2 - 7a 2 b 3 + 6tf 4 £ 4 . 

(<:) 1 -\- ax -\- a 2 -\- 2 a\ Written like this, the tf in the 
second term is understood as having 1 for an exponent; 
hence, if we represent the first term by a°, in value it will 
be equal to 1, since a = 1. (Art. 439.) Therefore, 1 
should be written as the first term when arranged according 
to the increasing powers of a. 

(178) tylWPc* = ± 2a % bc\ Ans. (Art. 521.) 
y — 32« 15 = — 2;z 3 . Ans. 

|/— l,728«V"jry = - \2a 2 d*xy\ Ans. 

(179) (a) (a-2x+±y)-(3z+2b-c). Ans. (Art. 408.) 

{b) — Zb — kc + d — (2/— 3e) becomes 
— [ob -f 4r — d -\- (2/— 3^)] when placed in brackets pre- 
ceded by a minus sign. Ans. (Art. 408.) 

(c) The subtraction of one expression or quantity from 
another, when none of the terms are alike, can be repre- 
sented only by combining the subtrahend with the minuend 
by means of the sign — . 



98 ALGEBRA. 

In this case, where we are to subtract 
2b — {3c -\-2d) — a from x, the result will be indicated by 
x - [2 b- {3c + 2d) -a.] Ans. (Art. 408.) 

( 1 80) (a) 2x z + 2x* + 2x - 2 
x - 1 



(*) 



2;r 4 + 2x 3 -f %x* 


— 2x 


- 2x* - 2x 2 


-2^ + 2 


2x< 


- 4* + 2 


x* — 4zax -j- <: 




2.r + # 





Ans. 



2^ 3 — Sax' -f 2rjr 



2-r 5 — lax' -f- 2<:.r — 4tf a ;r -f ac Ans. 

(<:) - a 3 + 3a*b-2b> 
5a 9 + 9#£ 



— ba b + 15# 4 £ - 10a 3 £ 3 

- 9a*b ' + 27tf 9 £ 3 - 18a£ 4 

— 5tf 6 + 6a*b - 10a 2 b 3 + 27tf 3 £ 3 - 18ab* 
Arranging the terms according to the decreasing powers 

of a, we have - 5a b + 6a*b + 27a 3 b* - 10a 2 b 3 - lSab\ Ans. 

(181) (a) kxyz The sum of the coefficients 

— 3xyz of the positive terms we find to 

— hxyz be + 13, since (+ 3) + (+ 6) + 
Gxyz (+ 4) = (+ 13). 

— §xyz When no sign is given before 
3xyz a quantity the + sign must al- 

_ ± xyz ^ns. ways be understood. The sum 
of the coefficients of the nega- 
tive terms we find to be — 17 since (— 9) + (— 5) -f- (— 3) = 
(— 17). Subtracting the lesser sum from the greater, and 
prefixing the sign of the greater sum ( — ) (Art. 390, rule 
II), we have (+ 13) + (— 17) = — 4. Since the terms are 
all alike, we have only to annex the common symbols xyz to 
— 4, thereby obtaining — kxyz for the result or sum. 



ALGEBRA. 99 

(b) 3a* + %ab + Ab 2 When adding polynomi- 
5a 2 — Sab + b 2 als, always place like terms 

__ a 2 + bab — b 2 under each other. (Art. 

ISa 2 - 20ab - 19b 2 393.) 

14# 2 — 3ab + 20b 2 The coefficient of a 2 in 

39tf 2 -24^ + 5£ 2 Ans. the result wil1 be 39 > since 

(+14) + (+18) + (-l),+ 

(_j_ 5) -j- (4- 3) — 39. When the coefficient of a term is not 

written, 1 is always understood to be its coefficient. (Art. 

359.) The coefficient of ab will be — 24, since (— 3) + 

(-20) + (+ 5) + (- 8) + (+ 2) = — 24. The coefficient of b 2 

will be ( + 20) + (- 19) + (-1) + (+!) + (+ 4) = +5. 

Hence, the result or sum is 39# 2 — %Aab + 5b 2 . 

(c) 4ctnn + Sab — Ac 

+ 2mn - Aab + Zx + dm 2 — 4/ 

6mn — ab — Ac + ?>x + dm 2 — 4/ Ans. 

(182) The reciprocal of 3. 1416 is— ---=.3183+. Ans. 
Reciprocal of .7854 = — ^— - = 1.273 +. Ans. 

5E5 < & ' Art - 481 -» 

x x — y 

(183) (a) ■ 1 — . If the denominator of the 

* ' x ' x — y y — x 

second fraction were written x — y, instead of y — x, then 
x — y would be the common denominator. 

By Art. 482, the signs of the denominator and the sign 

X y x V 

before the fraction — may be changed, giving — . 

We now have 

£ _ *—y _. x - xAr y = y A ^ 

x — y x — y x—y x—y 

x 2 X X 

(b) — = -I — — — . If we write the denominator 

v ' x 2 — 1 x + 1 1 — X 

of the third fraction x — 1 instead of 1 — x y x 2 — 1 will then 

be the common denominator. 



L.tfC. 



100 ALGEBRA. 

By Art. 482, the signs of the denominator and the sign 



x 



before the fraction may be changed, thereby giving 
We now have 

X* , x . x _x* + x{x-l) + x{x + l) 



X* _ 1 ■ x + 1 ' X — 1 X* — 1 

Ans. 



;tr 2 + ;r 2 - ;r + ^r 2 + * 3x* 



;tr 2 - 1 .r 2 - 1 

, . 3a — 4# 2* — b + £ , 13* — Ac , , - 

(c) - — ! J — , when reduced to 

a common denominator 

_ 12(3* - 4b) - 28(2a - b + c) + 7(13g - Ac) 

84 

Expanding the terms and removing the parentheses, we 
have 

36a - 48£ - 56a + 28b - 28c + 91a - 28<r 
84 
Combining like terms in the numerator, we have as the 

result, 

11a - 20b - 56c 



84 



Ans. 



(184) (a) 45^y°-90^y-360^-y= 
Ux A f (xy- 2x - 87). Ans. (Art. 452.) 

(b) a*b* + 2abcd+ c*d 2 = (ab + cd)\ Ans. (Art. 457.) 

(c) (a + by — (c—dy=(a + b + c — d)(a + b — c + d). 
Ans. (Art. 463.) 

(185) (a) If a man builds 20 rods of stone wall, and 
we consider this work as positive, or -f- , the work which he 
does in tearing it down maybe considered as negative, or — . 
If he tore down 10 rods, we could say that he built — 10 
rods. 

(b) See Arts. 388 and 398. 

tt ^„\ f \ 2ax -f- x* x x(2a4-x) a — x 

(186) (a) — s — —$--. =— K — — —rX • 

v ' v ' a 3 — x 3 a — x a' — x 3 x 

(Art. 502.) 



ALGEBRA. 101 

2# 4- x 



Canceling common factors, the result equals — 

a - 

a — x ) a 3 — x 3 ( a* -\- ax -\- x* 
a* — a 2 x 



ax -j- x v 
Ans. 



ax* — x s 
ax* — x* 

{b) Inverting the divisor and factoring, we have 

%n(%m*n — 1) (2m 2 n -f 1) (2/gg — 1) 

J^nfk — 1) {27u 2 u — 1) X dn 

Canceling common factors, we have 2?n*n -f- 1. Ans. 

9-r 2 - 4/ 



(c) 9 + y , -v- (3 + -5L_) simplified 

3* +2? 

x — y 

Qx* — 4y 2 x v 

Inverting the divisor, we have — 5 — X ~Tq ■ 

x — y ox —}— /C_y 

3 ^ 2y 

Canceling common factors, the result equals — — — — . Ans. 

(187) According to Art. 456, the trinomials 1 — 2x* 
-j- x* and 4,r 2 -f- 4.r -j- 1 are perfect squares. (See Art. 
458.) The remaining trinomials are not perfect squares, 
since they do not comply with the foregoing principles. 

04. 04. 

(188) (a) By Art. 481, the reciprocal of ^ = 1 -4- ^ 

= 1X 24 = 24- AnS ' 

{b) Since, by Art. 481, a number may be found from 
its reciprocal by dividing 1 by the reciprocal, the number 

= 1-5- 700 = .0014?-. Ans. 



102 ALGEBRA. 

(189) Applying the method of Art. 474, 
x+y 
x — y 

Sxy 



I2xy(x 7 -y), 2x*{x*+2xy-\-y*), 3y\x- y)\ 6(x*+xy) 



12xy (x - 


-y\ 2x\x+y), 


3y\x-y)\ 6x 


12xy 


%x\x+y\ 


Zy\x-y), 6x 


4, 


2x(x+y), 


y(x-y), 2 



2, x(*-\-y\ y{*-y\ i 

Whence, L. C. M. = (x + y) (x — y) 3 % y X 2 X % X 

*(x+y) xy(x~y) = nxy (x+y)* (x -y)\ 

(190) {a) 2 + la - 5a' - 6a' 

7a' 



lla* + 28a* - 35a" - 12a 6 Ans. (Art. 423.) 

(b) lx' - ly* + 6z* 

dxy 

Ylx'y — 12x*y* + lSx 9 yz 9 Ans. 

(c) 3b+5c- 2d 
6a 



ISab + SOac — \2ad Ans. 

(191) (a) See Arts. 359 and 361. 

(b) See Arts. 419 and 440. 

(c) See Art. 416. 

(192) (a) On removing the vinculum, we have 

2a - [3b + \lc - la - (2a + 2b) \ + {da - b - c) ]. 

(Art. 405.) 
Removing the parenthesis, 

2a - [3b + {Ic - la- 2a - 2b\ + {3a - b - c] ]. 
Removing the braces, 

2a - [3b + Ic - la - 2a - 2b + 3a - b - *]. 

(Art. 406.) 
Removing the brackets, 

2a - 3b - Ic + la + 2a -f 2£ - 3tf + £ +<f. 
Combining like terms, the result is ha — 3c. Ans. 



ALGEBRA. 103 

(b) Removing the parenthesis, we have 

la — [3a — {2a — 5a + ±a\]. 
Removing the brace, 

7a — [3a — 2a -J- ha — 4a\. 
Removing the brackets, 

la — 3a -f- 2a — 5a -f- Aa. 
Combining terms, the result is 5a. Ans. 

(c) Removing the parentheses, we have 

a — [2b + \3c — 3a — a— b\-\-\2a — b — c\\ 
Removing the braces, 

a— [2b -f 3c — 3a — a — b + 2a — b — c\. 
Removing the brackets, 

a — 2b — 3c + 3a + a -f b — 2a + b + c. 
Combining like terms, the result is 3a — 2c. Ans. 

(193) (a) (;r 3 +8) = (,r + 2) (.r 2 -2.r+4). Ans. 

(b) x 3 - 27 y = [x - 3y) (x 2 + 3xy +9/). Ans. 

(c) xm— nm -+- xy — ny = in (x —n) -\- y (x —n), 

or (x — n) (;;/ -f- y). Ans. 

(Arts. 466 and 468.) 

(194) Arrange the terms according to the decreasing 
powers of x. (Art. 523.) 

4x 4 + 8ax* + 4a? x* + 16&*x* + IQaPx + 16£ 4 (2x> + 2ax ■+■ 4b\ Ans. 
{2x*)' i =4x 4 . 



4x* + 2a x 



Sax 3 + 4a* x* 
8ax 3 + 4a* x* 



4x* + 4ax + 4b' 2 



I6^r 9 + 16ab*x + 16£ 4 
lSPx* + 16ab*x + 16b* 



/-•rAfirx c(a + b)-\-cd ac+bc-\-cd 

(195) , , V = ! — t4 — • Canceling c. which 

v ' (a-\- o)c ac -f be 

u* u a + d+'d , , ^ 

is common to each term, we have — != — = 1 4-- — — =. Ans. 

a-\- b ' a+b 



104 ALGEBRA. 

(196) (a) x+y+z- (x-y)-(y+z) - (-j)be v 
comes x -\- y -{- z — x -\-y — y — z -\-y on the removal of the 
parentheses. (Art. 405.) Combining like terms, 

x — x-\-y-\-y— y-\- y-\- z— z = 2y. Ans. 

(b) (2x -y + 4z)+(-x-y-4z)- (3x- 2y - z) be. 
comes 2x — y + 4z — x — y — 4z — 3x -f- 2y + z, on the 
removal of the parentheses. (Arts. 405 and 406.) . Com- 
bining like terms, 

2x — x — 3x — y — y -f- 2y -f 4z — 4z -f- z = z — 2x. Ans. 
(<:) a — \2a + (3« - 4a)] — 5a— \6a — [(7a+$a) -9a]\. 

In this expression we find aggregation marks of different 
shapes, thus, [, (,and {. In such cases look for the corre- 
sponding part (whatever may intervene), and all that is in- 
cluded between the two parts of each aggregation mark 
must be treated as directed by the sign before it (Arts. 405 
and 406), no attention being given to any of the other aggre- 
gation marks. It is always best to begin with the inner- 
most pair, and remove each pair of aggregation marks in 
order. First removing the parentheses, we have 

a— \2a-\-3a — 4a] — ha — \6a-[7a + 8a — 9a]}. 

Removing the brackets, we have 

a — 2a — 3a -\- 4a — 5a — {6a- 7a - 8tf + 9a\. 

Removing the brace, we have 

a — 2a — 3a + 4a — 5a — Qa -f 7a -f Sa — 9a. 

Combining like terms, the result is— 5a. Ans. 

(197) (a) A square x square, plus 2a cube b fifth, 
minus the quantity a plus b. 

(b) The cube root of x> plus y into the quantity a minus 
n square to the - power. 

o 

(c) The quantity m plus n, into the quantity m minus 
n squared into the quantity m minus the quotient of n 
divided by 2. 



ALGEBRA. 105 

(198) 

a* _ a i _ 2a - l)2a 6 - 4a 5 - 5a* + da 3 + 10a* + 7a + 2(2a 3 - 2a 2 - Sa - 2 
2a s _ 2a 5 - 4a* - 2a 3 Ans. 



-2a 5 
-2a 5 


- a* 
+ 2a* 


+ 5a 3 + 10a* 
+ 4a 3 + 2a? 




-Sa* 
-da* 


+ a 3 + 8a* + 7a 
+ da 3 + Qa* + Sa 






— 2a 3 + 2a* + 4a + 2 

— 2a 3 + 2a* + 4a + 2 



(199) (a) Factoring according to Art. 452, we have 
a*f (x 6 — 64). Factoring (x« — 64), according to Art. 463, 
we have 

(x 3 + 8) (x 3 - 8). 

Art. 466, rule, x 3 + 8 = (x + 2) (x* - 2x + 4). 

Art. 466, rule, x 3 - 8 = (x - 2) (.r 2 + 2x + 4). 

Therefore, ;r> 2 - 64.r 2 / = *y (x + 2) (^ 2 -2^+4) (*— 2) 
{x 2 + 2.r + 4), or x'f (x + 2) (x - 2) (** + 2x + 4) 
(^r 2 — 2.r+4). Ans. 

(£) # 2 — £ 2 — r 1 + 1 — 2tf + 2&\ Arrange as follows 
(Art. 408): 

(a 2 '- %a + 1) - (£ 2 - 2^ + *") = (a - l) 2 - (J - c)*. 
(Art. 455.) 

By Art. 463, we have 

(a - 1 + & - c) (a - I - [b -c]), 
or (a — l-{- b — c) (a—1 — b -\- c). Ans. 

(^) 1 — 16# 2 + Sac — c\ Placing the last three terms in 
parentheses (Art. 408), 1 — (16^ 2 — 8ac + c*). 

16a' - Sac +^ 2 = (4a - c)\ (Art. 455.) 

1 - (16tf 2 - Sac + c") = 1 - (4a - c)\ 

l.-(4a - 2 =[1 + (4a - c)] [l-(4a - c)]. (Art. 463.) 

Removing parentheses, and writing parentheses in place 
of the brackets, 

1 — (4a — cy = (1 + 4a — c) (1 — 4a + c). Ans. 

(200) See Art. 482. 



106 ALGEBRA. 

(201) The subtraction of one expression from another, 
if none of the terms are similar, may be represented only by 
connecting the subtrahend with the minuend by means of 
the sign — . Thus, it is required to subtract ha 3 b — la 2 b 2 -|- 
5ab 3 from a 4 — b\ the result will be represented by a* — b l — 
(5a 3 b — 7a 2 b 2 -f- 5ab 3 ), which, on removing the parentheses 
(Art. 405), becomes a'-tf - ba 3 b+1a 2 b 2 - bab\ From 
this result, subtract 3a* - ±a 3 b + Qa 2 b 2 + 5ab 3 - 3b\ 

a* — b" — ba 3 b -j- 7a 2 b 2 — hab 3 minuend. 
— 3a" + 3b A -\- 4:a 3 b — Ga 2 b' 7 ' — 5ab 3 subtrahend, with signs 
_ 2# 4 -f 2b" - a 3 b + a 2 b 2 - lOab 3 changed. (Art. 401.) 
Or, — 2a* — a 3 b -f a 2 b 2 — lOab 3 + 2b\ Answer arranged 
according to the decreasing powers of a. 

(202) (a) 3a - 2b -\-3c 3a - 2b + 3c 

2a — Sb — c becomes — 2a -f- Sb -f- c 

a + Gb + lc 
when the signs of the subtrahend are changed. Now, add- 
ing each term (with its sign changed) in the subtrahend to 
its corresponding term in the minuend, we have (— 2a) + 
(3a) = a; ( + Sb) + ( - 2b) = + 6b; (+ c) + (3c) = + 4<r. 
Hence, a -f- 6b -f- Ac equals the difference. Ans. 
(b) 2x 3 - 3x 2 y + 2xf 

x 3 -J- y 3 — xy* becomes 

2x 3 - 3x 2 y + 2xy 3 
— x 3 — y* + *? 



x * _ $ x y _j_ 2xy 3 - y 3 + .r/ 
when the signs of the subtrahend are changed. Adding 
each term in the subtrahend (with its sign changed) to its 
corresponding term in the minuend, we have x 3 — 3x 2 y -\~ 
2xy 3 — y 3 -J- xy 2 , which, arranged according to the decreasing 
powers of x, equals x 3 — 3x 2 y -f- xy' 2 -j- 2xy 3 —y 3 , Ans. 
(c) Ua + ±b-Qc — 3d 

11a — 2b + 4c — ±d 

On changing the sign of each term in the subtrahend, the 
problem becomes 



ALGEBRA. 107 

Ua + 4£ - 6c -3d 

-lla + 2b- ±c + 4:d 

3a + 6b - 10c + d 
Adding each term of the subtrahend (with the sign 
changed) to its corresponding term in the minuend, the dif- 
ference, or result, is 3a + 6b — 10c -f- d. Ans. 

(203) The numerical values of the following, when a 

= 16, b = 10, and x = 5, are: 

(a) (ab 2 x + 2abx) 4=a = (16 X 10 2 X 5 + 2 X 16 X 10 X 5) 
X 1 X 16. It must be remembered that when no sign is 
expressed between symbols or quantities, the sign of multi' 
plication is understood. 

(16 X 100 X 5 + 2 X 16 X 10 X 5) X 64 = (8,000 + 1,600) 
X 64 = 9,600 X 64 = 614,400. Ans. 

n\ o nr~ 2bx b — x _ r— 2X10X5. 

(b) 2y 4 a — ■ j + ■ ■ = 24/64 



a — b ' x y 16 — 10 ' 

10-5 ,„ 100 , , 96-100 + 6 2 1 

-— = 16-^ + 1 = = e=3- Ans ' 

(0 (3 - |/^) (x 3 - b*) {a' - b>) = (10 - |/16) (5 3 - 10 2 ) 
(16 2 - 10 2 ) = (10 - 4) (125 - 100) (256 - 100) = 6 X 25 X 
156 = 23,400. Ans. 

(204) (a) Dividing both numerator and denominator 
u ' i * 2 15 m xy* 1 . 

byl5 "^-754 r ^- Ans - 

tu\ x 2 -\ (j+1)(j-1) , 

\P) 1 — 7 — r~7T = ^~j — / , -.x when the numerator is 

v ' 4:x(x+ 1) ±x(x-\- 1) 

factored. 

Canceling (x -\- 1) from both the numerator and denom- 

x i 

inator (Art. 484), the result is . Ans. 

W (rt' -*')(«'-* /;+//) When faCt ° red b6COmeS 

(« + £) (*■ -ab + V) (a* + a b + /; 2 ) 

(a _ £) (^ + « b + j») (*» _ a b + ^)- t Art - ^>t>.) 



(?. (7. Z7.— 7 



108 ALGEBRA. 

Canceling the factors common to both the numerator and 
denominator, we have 

( a + b)(a* — ab + b 2 ) (a 2 -\-ab + b 2 ) _ a + b 
{ a -b){c? + ab-\- b 2 ) (a 2 -ab + b 2 )~ a - U 
x + b)a 3 + b 3 (a 2 -ab + b 2 a - b) a 3 - b 3 ( a * + ab + P 
a 3 + a 2 b a 3 — a 2 b 



- a 2 b + b 3 a 2 b - b 3 

a % b -ab 2 



ab 2 + b 3 ab 2 -b 

ab 2 + b 3 ab 2 -b 



(205) (a) 



1 + x 


-1+* 


1 


- x 2 


1 + * 


+ 1-X 



l-x 1 + x l — x A 2x 



1 1 1 + * + 1 — x l-x 2 

l-x^~l^-x l-x 2 

X 1 ~ X * = x. Ans. (See Art. 509.) 



l-x 2 l-x 2 " 2 
(b) of 1 a 3 + b 3 a 3 + 

!.S "T" " 



£ 3 a ab 3 ab 3 



a_ _ a-b a 2 b - b 2 (a — b) ~ d'b — ab 2 + b 3 ~ 

T 2 a~b~ ab 3 aT 3 ' 

a* + b* . a 2 b - ab 2 + b 3 _a 3 -\-b 3 ab 3 _a + 

X 



ab 3 ' ab 3 ab 3 N b{a 2 -ab+b 2 ) b ' 

Ans. 
(c) 1 _ 1 4 



1 , 3 - x Sx + 3 



Ans. 



1 . - y + 1 4 (Art. 509.) 

(206) -77 J r-: h- — 2 r-- If the denominator 

v J 2 — x 2 -\- x l x 2 — 4 

of the third fraction were written 4 — ;r 2 , instead of x 2 — 4, 
the common denominator would then be 4 — x 2 . 

D . ^ ^ 16x-x\ lQx-x 2 16x-x* 

By Art. 482, t ._ 4 - becomes - _ ^ 4 = - 4 _ ^ ■ 

„ 3 + 2;tr 2 - 3-r 16* - x 2 , , 

Hence, — — — =-, when reduced to a 

2 — x 2 -\- x 4 - x 2 

common denominator, becomes 



ALGEBRA. 109 

(3 4. 2*)(2 + *) - (2 - 3*)(2 - *) - (16* - * 3 ) _ 
4-* 2 
(6 -f- 7* + 2* 2 ) - (4 - 8* + 3* 2 ) - (16* - * 2 ) 
4 - x* 
Removing the parentheses (Art. 405), we have 
6 + 7* + 2* 2 - 4 + Sx - 3* 2 - 16* + ** 
4-* 2 
Combining like terms in the numerator, we have 

2 — x 



4-* 2 * 
Factoring the denominator by Art. 463, we have 

2-* 
(»■+*)(*-*)■ 

Canceling the common factor (2 — *), the result equals 

o-r-, or -^— . Ans. (Art. 373.) 



(207) (*) 

I o r 4*-4 ^ 5.r+10*'-4* + 4 = 1 0.r' + .g+4 Ang _ 
5* 5* 5* 

(Art. 504.) 

w ^t+t +1 ^^ 10 +rh Ans ' ( Art - 505 -) 

* + 4 ) 3* 2 + 2* + 1 ( 3x - 10 + 



3* 2 + 12* ' ^" l ~ 4 

- 10* + 1 

- 10* - 40 



41 
(c) Reducing, the problem becomes 

* 2 + 4* — 5 * — 7 



X 



x* 'x* — 8* + 7 

Factoring, we have 

lr + 5)(*-l) *_? 

..2 A 



(* - 1) (* - 7) 

*4- 5 
Canceling common factors, the result equals , . Ans. 



110 ALGEBRA. 

(208) (a) Writing the work as follows, and canceling 
common factors in both numerator and denominator (Arts. 
496 and 497), we have 

9//zV 5fq_ 24ry _ 

8/V X ~%xy X 90;«« ~~ 

9 X 5 X 24 X trfrfp^qx^y* _ Zmnxy 



8X2X90XWX«X/X^ 3 X^X; tyq 1 

(b) Factoring the numerators and denominators of the 
fraction (Art. 498), and writing the factors of the numer- 
ators over the factors of the denominators, we have 

(a — x) (a 2 -j- ax -f x 2 ) (a -f- x) (a -f- x) __ 
(a -j- x) {a 2 — ax -f- x 2 ) (a — x) (a — x) 

(a + x) {a' ±ax+ x 2 ) ^ A ^ 

(a — x) [a 2 — ax -j- x 2 )' 

(c) This problem may be written as follows, according to 
Art. 480 : 

3^4-4 



X 



a(dax -f 4) (Sax -j- 4)' 



Canceling a and (3ax-\-4:), we have- — — . Ans. 

daX — j— tfc 

(209) (a) — Imy ) Zom z y + 28;;z 2 / - Umy * 

-- bm 2 — Aiiny + 2j 2 

Ans. (Art. 442.) 

(b) a 4 ) ±a 4 - 3a b b - a*b 2 

4 — Sab — a 2 b 2 Ans. 

(c) Ax 2 ) 4.r 3 - Sx" + VIX 1 - 16-r 9 

x — 2x 3 + 3x b - 4.r 7 Ans. 

(210) (a) lQa 2 b*; a" + 4tf£; 4tf 2 - IQa'b + 5^ 6 + lax. 

(b) Since the terms are not alike, we can only indicate 
the sum, connecting the terms by their proper signs. 
(Art. 389.) 

(c) Multiplication : ±ac*d means 4 X a X c 1 X d. (Art. 
358.) 

, a 2 -f ^ -f ar ^ 2 -f ^ 2 - £ 2 - 2^ 

t^ 11 ) tf 2 + £ 2 -r 2 -2^ X tfY-tfc 4 

Arranging the terms, we have 



ALGEBRA. Ill 



a ' + ac + c z a 2 - lac + c* - P 

X 



a 2 — lab -f- b 2 — c 2 a"c — ac" 

which, being placed in parentheses, become 

a* + ac + c 2 (a 2 — lac + c 2 ) 



(a 2 - lab + b 2 ) - c 2 a*c - ac" 

By Art. 456, we know that a 2 — lab -f b 2 , also a 2 — lac 

-f- c 2 , are perfect squares, and may be written (a — b) 2 and 

(a-c)\ 

Factoring a*c — ac* by Case I, Art. 452, we have 

a* -f ac -f ^ 2 (g - r) 2 -b 2 _ 

(a — b) 2 — c 2 ac (a 3 — c*) 

a 2 -\- ac -\- c 2 (a — c — b) (a — c ~\- b) 

X 



{a — b — c) {a — b -\- c) ac (a — c) (a 2 -f- ac -\- c 2 )' 

(Arts. 463 and 466.) 

Canceling common factors and multiplying, we have 

a — c 4- b a -\- b — c 

or — - t 1—, — r-7 -. Ans. 



(a — b -\- c) ac {a — <:)' ac (a — b -\- c) [a — c)' 

(212) The square root of the fraction a plus b plus £ 
divided by n, plus the square root of a, plus the fraction b 
plus <: divided by n y plus the square root of a plus £, plus 
the fraction c divided by n, plus the quantity a plus b, into 
r, plus a plus for. 

(213) (a)— BT- + HP. 

We will first reduce the fractions to a common denomina- 
tor. The L. C. M. of the denominator is 6(Xr 2 , since this is 
the smallest quantity that each denominator will divide 
without a remainder. Dividing 60x 2 by 3, the first denom- 
inator, the quotient is 2(U' 2 ; dividing QOx* by 5x, the second 
denominator, the quotient is 12^r; dividing QOx 2 by 12;r a , the 
third denominator, the quotient is 5. Multiplying the cor- 
responding numerators by these respective quotients, we 
obtain 20-r 2 (4r -j- 5) for the first new numerator ; 12x(3x — 7) 
for the second new numerator, and 5 X 9 = 45 for the third 
new numerator. Placing these new numerators over the 
common denominator and expanding the terms, we have 



112 ALGEBRA. 

20.r 2 (4.r+5)-12.r(3.r-7)+4o_80^ 3 +100^-3C.r a +84.r+45 
6(Lr 2 ~~ 60-r 2 

Collecting like terms, the result is 

SOx 3 + 64jt 2 -f 84.r + 45 . 

60? ' AnS * 

(b) In — -. — ■ — r 4- —7 r, the L. C. M. of the denomi- 

v ' 2a(a-{-x) "Za{a — xy 

tiators is 2a(a 2 — x" 2 ), since this is the smallest quantity that 
each denominator will divide without a remainder. Dividing 
2a(a* — x 2 ) by 2a(a -f- x), the first denominator, we will 
have a — x; dividing 2a(a* — x' 2 ) by 2a(a — x), the second 
denominator, we have a -f- x. Multiplying the correspond- 
ing numerators by these respective quotients, we have 
(a — x) for the first new numerator, and (a -f- x) for the 
second new numerator. Arranging the work as follows: 

IX (a — x) = a — x = 1st numerator. 

1 X (a + x) = a -\- x = 2d numerator. 

or 2a = the sum of the numerators. 

Placing the 2a over the common denominator 2a(a i — x 7 ), 
we find the value of the fraction to be 
2a 1 



2a(a*—x*) a 2 - x v 



Ans. 



y x ~\-y x * + x y y x ~\-y x ~\-y 

The common denominator = y (x -j- y). Reducing the 
fractions to a common denominator, we have 

x { x +y)+y 2 + x y = x* + 2xy+y 2 _ x+y Ang 

y(x+y) y( x +y) y 

(214) (a) Apply the method of Art. 474 : 



6ax 


ISax 2 , 72ay 2 , 


12xy 


2y 


dx, 12y\ 


2y 


3 


3x, 6y, 


1 



x, 2y, 1 
Whence, <5ax X 2; x 3 x ^ X 2j = 72ax*y\ Ans. 



4(1 + 


*). 


4(1- 


-*), 


2(1- 


-**) 


2, 




2(1- 


-*), 


1- 


- X 



(a — b) (b—c), (b—c) (c — a), (c—a) (a- 


-b) 


(b—c), (b — c) (c—a), (c—a) 




1, c—a, c—a 





(b) 2(1 + x) 
2(1 -x) 

""l, 1, 1 

Hence, L. C. M. = 2(1 + x) X 2(1 - x) = 4(1 - x"). Ans. 

(c) a—b 
b-c 
c—a 

1, 1, 1 

Hence, L. C. M. = (# — 3) (b — c) (c — a). Ans. 

(21 5) 3.r 6 - 3 -\-a -ax" = (3 - «)^ - 3 + <z = 

(3 — a) (x e — 1). Regarding x 6 — 1 as (;r 3 ) 2 — 1, we have, by 
Art. 462, x 6 - 1 = (x 3 ) 2 - 1 = (x 3 - 1) (x* + 1). .r 3 - 1 = 
(x - 1) (;tr 2 + x + 1) ; x z + 1 = (jut + 1) (^ a - * + 1). Art. 
466. 

Hence, the factors are 

(*»+*.+ !) („ r 2 - * + 1) (* + 1) (* - 1) (3 - a). Ans. 

(216) Arranging the terms according to the decreasing 
powers of x, and extracting the square root, we have 

** + x*y + 4i*y + 2^7 3 + 4/ (;tr 2 + fcry + 2y\ Ans. 
;r 4 



2* 2 + \xy 



x*y + 4J^ry 



2^ 3 + xy+ 2/ 



4*y + 2.t'7 3 + 4/ 

4*y + 2*y + 4/ 



(217) The arithmetic ratio of x 4 — 1 to x + 1 is^r* — 1 — < 
(*+l) =^ 4 -^-2. Art. 381. 

^ - 1 



The geometric ratio of x A — 1 to x + 1 is 
H- * - 1 == (** ^ 1) (x — 1). Ans. 



*4-l 



ALGEBRA. 

(QUESTIONS 218-257.) 



(218) (#) According to Art. 528, x* expressed radi- 
cally is 4/^; 

3x*y* expressed radically is 3 \/xy- 3 ; 

3x h j V = 3fjp7, since z l = z % . Ans. 

^ i i *v 

^ + «:' (a + b) + m - n b* ' 

(c) \^x 6 = x\ Ans. \HP^x % . Ans. 
(^^V = {b l x h Y = b^x % . Ans. 

(219) 3|/S = /l89. Ans. (Art. 542.) 

cfb^Wc = 4/ a 4 b 5 c . Ans. 2^^ = ^32^ 6 . Ans. 

(220) Let x = the length of the post. 

x 
Then, — = the amount in the earth, 
o 

3x 

— = the amount in the water. 

f + ^ + 13 = ^. 

7^+15^-4-455 = 35^-. 

- 13* = - 455. 

x = 35 feet. Ans. 



< 221 > '" ^,+ »V, • 



In order to transform this formula so that / Q may stand 
alone in the first member, we must first clear of fractions. 
Clearing of fractions, we have 

For notice of the copyright, see page immediately following the title page. 



116 ALGEBRA. 

tW x s x + tW % s % = W x s x t x +W tt% tr 

Transposing, we have 

- W % s, t, = W x sj.-t W 1 s t - t W % s r 
Factoring (Arts. 452 and 408), we have 

whence , , i = (^+y/^M, Ans . 

(222) Let x = number of miles he traveled per hour. 

48 
Then, — ■ = time it took him. 

x 

48 

= time it would take him if he traveled 4 



miles more per hour. 
In the latter case the time would have been 6 hours less ; 
whence, the equation 

48 48 
x-\- 4 x 
Clearing of fractions, 

48* = 48* + 192 - 6* a - 24*. 
Combining like terms and transposing, 

6* 2 + 24* = 192. 
Dividing by 6, * 2 + 4* = 32. 

Completing the square, * 2 -f- 4* + 4 = 36. 
Extracting square root, x + 2 = ± 6 ; 

whence, x ■= — 2 + 6 = 4, or the 

number of miles he traveled per hour. Ans. 



(223) W S = ^4^=<T CPD 



/( 2 +5) " 2/ + 






Cubing both members to remove the radical, 

CPD % 



5 3 = 



v+!Z 



CPD*d % 
Simplifying the result, S 3 = a . 



ALGEBRA. 117 

Clearing of fractions, 

2 S a fd 2 + S z fD* = CPB 2 d\ 

Transposing, CPD 2 d 2 = 2 S 3 fd* + S'fD" ; 

„ 2 5 yd 2 + ^ 3 /^ 2 (^ 2 + £> 2 ) /5 3 . 
wh ence, P= ^^ ■ = ^-^ . Ans. 

(b) Substituting the values of the letters in the given 
formula, we have 

p _ (2 X 18 2 + 30 2 ) X 864 x'6 3 _ (648 + 900) X 864 X 216 = 
10 X 30 2 X 18 2 9,000X324 

288,893,952 .. . . . 

2,916,000 =»0-l> nearly. Ans. 

(224) (a) 3x + 6 — 2* = 7*. Transposing 6 to the 
second member, and 7* to the first member (Art. 561), 

3x — 2* — 7* = — 6. 
Combining like terms, — 6x = — 6 ; 

whence, x = 1. Ans. 
(b) 5x - (3x - 7) = Ax - (6x - 35). 

Removing the parentheses (Art. 405), 

_5x — 3x + 7 = Ax — 6* + 35. 
Transposing 7 to the second member, and 4^- and — 6x to 
the first member, 5x — 3x — Ax -\- 6x = 35 — 7. (Art. 561.) 
Combining like terms, Ax = 28 ; 

whence, * = 28 -f- 4 = 7. Ans. 
(0 (x+5Y-(A-xy = 21x. 

Performing the operations indicated, the equation becomes 

x* + 10* + 25 - 16 + 8x - x 2 = 21* 
Transposing, x* — x* + 10* + 8* — 21* =16 — 25. 
Combining like terms, — 3* = — 9. 

Dividing by — 3, x = 3. Ans. 

(225) (a) Simplifying by Art. 538, 

|/~27"= 4/T X 4/3 = 34/3. 
2V^48" = 24/I6 X /3 - 84/3. 

3/108 = 34/36 x 4/3 = 184/a! 

Sum = 294/3. Ans. (Art. 544.) 



118 ALGEBRA. 

(b) ^T2S = ^~6rx^2= 4^2. 
f^S6 = ^Md X^2 = 7^2". 
f r W=f r ~S~X0 = 2^2. 
Sum = 13^2. Ans. (Art. 544.) 



w 



\ / 1=\ / 1^1=\ / JrA^' (Art. 540.) 



16 4 



r 6 r 6 6 r 36 6 r 

r 27 r 27 3 f 81 9 r 
Sum=(I + L+l.) / e = g./6. An, 

(226) Let * = the capacity. 
Then, x — 42 = amount held at first ; 

7(*-42) = *; 
7*-294 = *; 
6*= 294; 
;r = 49 gallons. Ans. 

(227) (a) 2 4/3jt + 4 - * = 4. 

Transposing, Art. 579, so that the radical stands alone 
in the first member, 2 \/3x -f 4 = * + 4. 

Squaring both members, since the index of the radical 
is understood to be 2, 4(3* -J- 4) = (* -f- 4) 2 , 

or 12* + 16 = x 2 + 8* + 16. 

Transposing and uniting terms, 

- x 2 - Sx + 12* = 16 - 16. 

_ 3? _|_ 4^ = o. 

Dividing by — *, x — 4 = ; 

whence, x = 4. Ans. 

(£) |/3*- 2 = 2(*- 4). 

Squaring, 3* - 2 = 4(* - 4) 2 , 

or 3* - 2 = 4* a - 32* + 64. 



ALGEBRA. 119 



Transposing, — 4* 3 + 32* -f 3* = 64 + 2. 
Combining terms, — 4* 2 -j- 35* = 66. 

Dividing by — 4, * -— = —. 

Completing the square, 

a 35* , /35\ 2 66 . 1,225 



+ 



(?) 



4 ' \8/ 4 ' 64 

35* , /35V 1,056 . 1,225 169 



(f)'- 1 



^-T-+ IT =-^F + 



64 ' 64 64 



Extracting the square root, 



35 , 13 
Transposing, * = —• ± — - = 6, or 2 T . Ans. 

o o 4 

(^) */* + 16 = 2 + y* becomes * + 16 = 4 + 4y* + *, 
when squared. Canceling * (Art. 562), and transposing, 

— 4|/* =4-16. 

- 4/r = -- 12. 

V* = 3; 
whence, x = 3 3 = 9. Ans. 



(228) («) /3J^5=^-±^£ 

Clearing of fractions, 



*|/3* - 5 = 4/7.r 2 + 36*. 
Removing radicals by squaring, 

* 2 (3* - 5) = 7* 2 + 36*. 
3* 3 - 5* 2 = 7* 2 + 36* 
Dividing by *, 3* 2 — 5* = 7* + 36. 
Transposing and uniting, 

3* 2 - 12* = 36. 
Dividing by 3, * 2 — 4* = 12. 



120 ALGEBRA. 

Completing the square, 

x* - ±x + 4 = 16. 
Extracting the square root, 

x—2 = ± 4; 
whence, x = 6, or — 2. Ans. 

(b) ' x* — (b — a)c = ax — bx -j- <r;r. 

Transposing, x* — ax -\- bx — ex — (b — a)c. 
Factoring, x*— (a — b + ^)-r = be — ae. 

Regarding (a— b-\- c) as the coefficient of x, and com* 
pleting the square, 

a* - 2ab + U 1 - 2ac -f %bc_ + ^ 2 . 

4 
a — b -\- c a — b — c 

x — = ± ^Y— 

2a — 2b 2c 

x = a — b, or c. Ans. 
(c) (x — 2) (x — 4) — 2 (;r — 1) (.*• — 3) = 0, becomes 

x* — 6x -j- 8 — 2.r 2 + 8^" — 6 = 0, when expanded. 
Transposing and uniting terms, 

- x* + 2.r = - 2. 
Changing signs, x" 1 — 2.r = 2. 

Completing the square, x* — 2x -f- 1 = 3. 
Extracting the square root, x — 1 = ± \Z 1 6\ 

whence, x = 1 ± 4/3? Ans. 

(229) (a) ^T^^b= {a + b) ^- b \ 

\/X 

Expanding and clearing of fractions, 

yV - 4abx = a* - b\ 
Squaring both members, 

x* - ±abx = a 4 - 2a*b* + b\ 



ALGEBRA. 121 

Completing the square, 

x* - ±abx + ^a'b 1 = a' + 2a 2 b* + b\ 
x-2ab = ± (a 2 + P). 

x = (a* + 2ab + b*), 

or- ( a 2 -2ab+P). 
x=(a + b)\ or -{a — b)\ Ans. 

becomes 




— yx — 1 + yx -f 1 = 1 when cleared of fractions. 
Squaring, 

x — 1 — 2^r 2 - 1 + * + 1 = 1. 

— 2^ 2 - 1 = 1 - 2x. 
Squaring again, 4.r 2 — 4 = 1 — 4;r -f- 4ff*. 
Canceling 4;r 2 and transposing, 

4* = 5. 

;r = — = 1 T Ans. 
4 4 

(230) &r- 2^ = 51. (1) 

19* - 3j/ = 180. (2) 
We will first find the value of x by transposing — 2y to the 
second member of equation (1), whence 5x = 51 + 2/» and 

^£l±^. (3) 



This gives the value of x in terms of y. Substituting the 
value of x for the x in (2), (Art. 609.) 
19(51 + ^) _ 3/=18a 

Expanding , 3 969 + 38^_ 37 = 18a 



Clearing of fractions, 969 + 38 y — 15y = 900. 
Transposing and uniting, 23 y = — 69. 

y = — 3. Ans. 
Substituting this value in equation (3), we have 

-r = 5i^ = 9. Ans. 



122 


ALGEBRA. 


(231) (a) 


2^ -27* = 14. 




--¥='■ 


X*- 


".*+0" -'+©"- * 




27 , 29 
4 4 




56 -m 

* = T = 14, 




2 1 
or *=__ = __. 


Hence, 


jr = 14, or — — . Ads 


(*) 


"-X + l2 = - 


Transposing, 


3 2x 1 



Completing the square, 
x 



3 ^\3/ 12^9 36 



Extracting the square root, 

Transposing, x — ^ + - = -, 

111 

Therefore, .ar = - or -. Ans. 

<i o 

(c) x*-\-ax = bx-\-ab. 

Transposing and factoring, 

x* -f- (a — b) x = a b. 

*■+(— *)*+( !L r ? )=-*+( !L J i )- 

4a£-f-a 2 -2a£+£ 2 _ ^ + 2a£ + £ a 
4 ~~ 4 



ALGEBRA. 123 



Extracting square root, 



a — b _ a-\-b 



1 2 ^2 

a — b , a-\- b . 

a — b a 4- b 
or x= - Y-=~ a - 

Therefore, x = b, or — a. Ans. 

(232) Let x = rate of current. 

y = rate of rowing. 
Down stream, the rowers are aided by the current, so 
r + y=*l2. 

Since it taices them twice as long to row a given distance 

up stream as it does down stream, they will go only - as far 

1 l 

in 1 hour, or — of 12 — 6 miles per hour up stream. 

x+y=Vl. (1) 
-x+y= 6. (2) 



Subtracting, %x = 6, and x = 3 miles per hour. 

Ans. 

(233) («)i^tl-^=10(,-l). 

Reducing the last member to a simpler form, the equation 
becomes 

o Z 

Clearing of fractions by multiplying each term of both 
members by 6, the L. C. M. of the denominators, and chang- 
ing the sign of each term of the numerator of the second 
fraction, since it is preceded by the minus sign (Art. 567), 
we have 

20-r + 6 - 18* + 21 = QOx - 60. 

Transposing terms, 20* — 18* — QOx — — 60 — 21 — 6. 

Combining like terms, — 58* = — 87. 

Changing signs, 58* = 87 ; 

87 1 
whence, x = — = 1 -. Ans. 

a. a. iv.— 8 



124 ALGEBRA. 

(b) (a* + *Y = x* = W + a\ 

Performing the operation indicated in the first member, the 
equation becomes 

a * _]_ % a *x + x* = x* + W -f a\ 
Canceling x* (Art. 562) and transposing, 

Icfx = 4« 2 + a' — a\ 
Combining like terms, %a % x = 4# 2 . 
Dividing by 2<z 2 , x = 2. Ans. 

'^ * - 2 x + 2 "~ ^ 2 - 4* 

Clearing of fractions, the equation becomes 

(x - 1) {x + 2) - (x + 1) (* - 2) = 3. 
Expanding, x * -\- x — 2 — ^ a + ^4-2 = 3. 
Uniting terms, 2;r = 3. 

* = | = 4 Ans. 

(234) ll*+3y=100. (1) 

4jt _ 7 7 = 4. (2) 

Since the signs of the terms containing x in each equation 
are alike, x may be eliminated by subtraction. If the first 
equation be multiplied by 4, and the second by 11, the co- 
efficients in each case will become equal. Hence, 
Multiplying (1) by 4, 44^ + 12y = 400. (3) 

Multiplying (2) by 11, 44^ - 77/ = 44. (4) 

Subtracting (4) from (3), 89/ = 356. 

y = 4. Ans. 
Substituting this value for/ in (2), 

4* - 28 = 4. 
4.x = 32. 
x — 8. Ans. 

(235) (a) y* = 243. 
Extracting fifth root of both terms, 

y$=3. 
Cubing both terms, y = 3 3 = 27. Ans, 



ALGEBRA. 125 



(b) x 10 + 31x b - 10 = 22, 




or x 10 + 3Lr 6 = 32. 




Completing the square, 




-+«>.+ ©"-"■+ (¥)• 




«• + ,*• + (?)'-»+ »? = 


1,089 
4 


31 33 
Extracting square root, ^ & + — = ± — . 





m • 31 , 33 2 

Transposing, *V== __ + _=_ or 1. 

31_33__64__ 
* ~ "2 2~ 2~ **' 
whence, jf = ^7T= 1 

or * = f^32~= - 2. Ans. (Art. 600.) 

(c) x 3 - 4-r f = 96. 

Completing the square, 

x* — ±x l + 4 = 96 + 4 = 100. 
Extracting square root, P — 2 = ± 10. 
Transposing and combining, x* = 12, or — 8. 
But, x l — j/P = 12, or — 8. 

Removing the radical, ;r 3 = 144, or (— 8) 2 . 

4/T4I = #8 X 4^18 = 2^18. (Art. 538.) 
fp 5) r =(-8)« 
Hence, ^ = 2fl8, or (- 8) § . Ans. 

(236) (a) The value of a is the same as 1. (Arts. 
438 and 439.) 

(b) -^i = a. Ans. (Art. 530.) 

(c) y(3x* + 5xy + Gxyy = 3.r 2 + hxf + 6*> = 3 X 2 2 + S 
X 2 X 4 3 -f 6 X 2 2 X 4 = 12 4- 640 + 96 = 748, when x — 2. 
and y = 4. Ans. 



126 ALGEBRA. 

,~~-x t n 6*+l 2* -4 2* - 1 

(237) (a) _±™— _ = __ 

Clearing of fractions, 

(6* + 1) (7* - 16) - 15(2* - 4) = 3(2* - 1) (7*- 16) 
Removing parentheses and expanding, 

42* 2 - 89* - 16 - 30* + 60 = 42* 2 - 117* + 48. 
Canceling 42* 3 (Art. 562) and transposing, 

117* - 89* - 30* = 16 - 60 + 48, 
Combining terms, — 2* = 4. 

* = — 2. Ans. 

... ax 3 .ax 

(*) F^XF + a + T =0 - 

abx* -f- #^ — ## 2 * -f- <^* — ##* 2 = 0. 
Transposing and uniting, acx — ab 2 x = — abc. 

a(c — b 2 )x = — abc. 

abc 



a(c - d*)' 
Canceling the common factor a and changing two of the 
signs of the fraction (Art. 482), 

be 





*=r- 


— . Ans. 
■ c 




(<) 


\/x— 3 


4/* — 4 






4^+7" 


~ 4^+1 




Clearing of ffactions, 






Wi- 


- 3) ( 4 /^+l) = 


(V^ + 7)(V5- 


-4)- 




x — 2 4/* — 3 = 


* + 3 \/x - 28. 




Transposing 


• and canceling 


* (Art. 562), 




— 


24/*- 34/* = 


3-28. 






— 04/* = 


- 25. 






^= 


5. 






* = 


25. Ans. 





ALGEBRA. 127 

(238) (a) j/l by Art. 540 = \^\ = ^\ = \^' 

Ans. 



W Ti V 7 ~u y 7 X 7 - n X 7 ^ 7 - 77 ^ 7 - Ans - 



s /2* A */>Zx z 2 z 3/ 



(c) zy — = zy — - x — j = - V% xz * — |/2*s- a . Ans. 
z z z z 



When the denominators contain both simple and com- 
pound expressions, it is best to remove the simple expressions 
first, and then remove each compound expression in order. 
Then, after each multiplication, the result should be reduced 
to the simplest form. 

Multiplying both sides by 36, 

5* — 4 ' 

144* - 432 OA 

or — — = 20. 

5* — 4 

Clearing of fractions, 

144* - 432 = 100* - 80. 

Transposing and combining, 

44*= 352; 

whence, * = 8. Ans. 

3^ fox \ 

(b) ax = — becomes, when cleared of fractions, 

2ax — Sa -f- bx = 1. 
Transposing and uniting terms, 

lax + bx = 3a + 1. 
Factoring, (2a + b)x = Ba + 1 ; 

whence, * = J , . . Ans. 
2a -J- b 



128 ALGEBRA. 

d £ x 

(c) am — b '-, — h — = 0, when cleared of fractions — 

v ' b in 

abn? — b 2 m — amx -f- bx = 0. 

Transposing, bx — amx = b'in — abm 2 . 

Factoring, (b — am)x = bm(b — am) ; 

t bm(b — am) . . 

whence, x = — jt +- = bin. Ans 

[b — am) 



(240) x-\.y=.YS. 




(i) 


xy = 36. 




(2) 


Squaring (1) we have 






x* + 2x)>-\-f = 1G9. 




(3) 


Multiplying (2) by 4, 4xy = 144. 




(*) 


Subtracting (4) from (3), 






,r 2 - 2xj'+f = 2o. 




(5) 


Extracting the square root of (5), 






x — y — ± 5. 




(6) 


Adding (6) and (1), 2.r = IS oi 


•8, 




x = 9 or 


4. 


Ans. 


Substituting the value of x in (1), 






9+/ = 13, 






or 4+ y =13; 






whence, y = 4, ) 


Ans. 



or y = 9. ) 

(241) **--y = 98. (1) 

x-y = 2. (2) 

From (2), x=%+y. (3) 

Substituting the value of x in (1), 

8 + 12j'+6j a +J s -J' 3 = 98. 
Combining and transposing. 

6/ + Uy = 90. 
f + 2j' = 15. 
y + 2j'+ 1 = 15 + 1 = 16. 
.7+1= ±4. 

j' = 3, or — 5. Ans. 
Substituting the value of y in (,3), :r = 5, or — 3. Ans 



ALGEBRA. 120 



(242) Let x = the whole quantity. 

2x 
Then, — + 10 = the quantity of niter, 
o 

X 1 

— — 4~r = the quantity of sulphur. 



^■(~ + 10 )— 2 = the quantity of charcoal. 

Hence, :* = *— + 10 + g- - % + ;*r ( T +10 j- 2. 

Clearing of fractions and expanding terms, 

42;r = 28# + 420 + 1x - 189 + 4# + 60 - 84. 
Transposing, 

42^- - 28# - 7# — 4# = 420 — 189 + 60 — 84. 
3x = 207. 
#=69 lb. Ans. 

(243) Let x = number of revolutions of hind wheel. 
Then, 51 + x = number of revolutions of fore wheel. 

Since, in making these revolutions both wheels traveled 
the same distance, we have 

16x = 14 (51 + x). 
16x = 714 + Ux. 
2x = 714. 
x= 357. 

Since the hind wheel made 357 revolutions, and since the 
distance traveled for each revolution is equal to the circum- 
ference of the wheel, or 16 feet, the whole distance traveled 
= 357 X 16 ft. = 5,712 feet. Ans. 

(244) (a) Transposing, 

5x* - 2x* = 24 + 9. 
Uniting terms, 3# a = 33. 

x> = 11. 
Extracting the square root of both members, 

x — ± |/IT. Ans. 



130 ALGEBRA. 

(h , J L'-Z. 

W 4;tr 2 6;r* ""3" 

Clearing of fractions, 9 — 2 = 28^*. 
Transposing terms, 28-r* = 7. 

Extracting the square root of both members, 

x = ± — . Ans. 

f! _ *' ~" 10 - 7 _ 5Q + ;tra 

{C) 5 15 ~ 7 25 ' 

Clearing of fractions by multiplying each term of both 
members by 75, the L. C. M. of the denominators, and ex- 
panding, 

Ihx* - 5x* + 50 = 525 - 150 - Zx\ 

Transposing and uniting terms, 
13-r 2 = 325. 



Dividing by 


13, 


or x = ± 5. Ans. 




(345) 
From (1), 




4* + 3j/ = 48. 
- 3^r + 5j/ = 22. 

48 - 4-r 
^ = 3 ' 


(1) 
(2) 

(3) 


From (2), 




22+ dx 


(4) 


Placing (3) ; 


md 


(4) equal to each other, 
48 - ix _ 22 + %x 





3 5 

Clearing of fractions, 

240 — 20^ = 66 + 9x. 

Transposing and uniting terms, 

- 29x = - 174, 

or x = 6. Ans. 

Substituting this value in (4), 

22+18 . . 
y = — -£ — = 8. Ans. 



ALGEBRA. 



131 



(246) Let * = speed of one. 
* -f- 10 = speed of other. 



Then, 



1,200 



x 
1,200 
* + 10 



= number of hours one train required. 



= number of hours other train required. 



1,200 1,200 



10. 



x x + 10 ■ 

1,200* + 12,000 = 1,200* + 10* a + 100*. 
- 10* a - 100* = - 12,000. 
10* 2 + 100* = 12,000. 
* a + 10* = 1,200. 
x 2 + 10* + 25 = 1,200 + 25 = 1,225. 
x+5= ±35. 



*+10 



* = 30 miles per hour. ) . 
= 40 miles per hour. J 



(247) 2* - 
3y + 



y 



5 

* — 2 



-4 = 



9 = 



cleared of fractions, becomes 



10*-j/ + 3-20 = 0. (1) 

9y + * - 2 - 27 = 0. (2) 

Transposing and uniting, 10x—y = 17. (3) 

x + 9/ = 29. (4) 

Multiplying (4) by 10 and subtracting (3) from the result, 

10* +90? = 290 
10*-- y = 17 

91y = 273 
y = 3. Ans. 
Substituting value of y in (4), 

*+ 27 = 29. 

* = 2. Ans. 



(248) (a) ffi X VI = 2 J X 3* (Art. 547.) 

2* X 3* = W X 1 i/¥= X ¥'M X 27 = 4 5 /8G4. Ans. 



132 ALGEBRA. 

{b) 4^7 X Va^= {Zaxf X (ax 1 ) 1 = y'sZFx Va^x 1 = 

i/Sa'x 11 . Ans. 
(c) %*/x~y X 3V*y = %X %(xyf X {x'y) 1 = G'^yT Ans. 

(249) Let x = the part of the work which they all can 
do in 1 day when working together. 

rru • 1 ^ 1.1,2 

Then, since - = -, - + - + _=*; 

or, clearing of fractions and adding, 

15 = 30x, and x — —. 

Since they can do — the work in 1 day, they can do all 
of the work in 2 days. Ans. 

(250) Let x == value of first horse. 

y = value of second horse. 
If the saddle be put on the first horse, its value will be 
x -\- 10. This value is double that of the second horse, or 2y, 
whence the equation, x -\- 10 = 2y. 

If the saddle be put on the second horse, its value isy + 10. 
This value is $13 less than the first, or x — 13, whence the 
equation, y + 10 = x — 13. 

x + 10 = %y. (1) 

y + 10 = x - 13. (2) 

Transposing, x — %y = — 10. (3) 

_ x _j_ y - _ 23. (4) 

Adding (3) and (4), — y = — 33. 

y = $33, or value of second horse. Ans. 
Substituting in (1), x + 10 = 66 ; 

or x = $56, or value of first horse. Ans. 

(251) Let x = A's money. 

y = B's money. 

If A should give B $5, A would have x — 5, and B, y -\- 5. 
B would then have $6 more than A, whence the equation, 



ALGEBRA. 133 

> + «-(*- 6) = 6. (1) 

But if A received $5 from B, A would have x + 5, and B, 
r — 5, and 3 times his money, or 3 (x -f- 5), would be $20 
more than 4 times B's, or 4 ( _y — 5), whence the equation, 
S(*+5) -4(^-5) = 20. (2) 

Expanding equations (1) and (2), 

y -f 5 - ; * + 5 = 6. (3) 

3 * +15 - 4j/ + 20 = 20. (4) 
Transposing and combining, 

y — x = — 4. (5) 

- Ay + 3.*r = - 15. (6) 
Multiplying (5) by 4, and adding to (6), 

Ay — Ax = — 16. 

— 4y + 3* = — 15. 

- ;tr = - 31. 
* = 31. 

Substituting value of x in (5), 

y - 31 = - 4. 

j/ =27. 
Hence, ;r = $31, A's money 

j = $27, B's money 



Ans. 



(252) (a) x*-q x = 16. 

Completing the square (Art. 597), 

x* - 6x + 9 = 16 + 9. 
Extracting the square root, x — 3 = ±5. 
Transposing, x = 8, or — 2. Ans. 



(b) x* - 7-r = 8. 



81 



whence, ;r = 8, or — 1. Ans. 



134 ALGEBRA. 

(c) 9x' - l%x = 21. 

tv -a- u q i 12x 21 
Dividing by 9, x — = — , 

, 4* 7 



Completing the square, 

3 



±x /2 y _ 7 4 __ 25 



2 5 

Extracting square root, ;r — — = ± — . 

o o 

Transposing, ;r = - + - = -, 

2 5 3 

° r " = 3-3^-3 = - L 

7 
Therefore, x = -, or — 1. Ans. 

(253) (*-!)-* = **. Ans. (Art. 526, III.) 

(7«|/«*)-* = nt-l(ni)-*-= m-*n * = — y-y. Ans. 

(^ 2 )^ = ^i^-f, or I^T" 2 , or f-^-. Ans. (Art. 530.) 

(254) 

Let x = number of quarts of 90-cent wine in the mixture. 
y = number of quarts of 50-cent wine in the mixture 
Then, x +y=Q>0, (1) 

and 90*+50j>=4,50b=75X 60. (2) 
Multiplying (1) by 50, 

50.r+50j/= 3,000. (3) 

Subtracting (3) from (2), 

40*= 1,500; 

whence, x = 37— qt. Ans. 

Multiplying (1) by 90, 90* + 90/ = 5,400 (4) 
Subtracting (2), 90*+50j/ = 4,500 (2) 

40y= 900; 

whence, y = 22— qt. Ans. 

6 



6 


3 


6y — 10 




5 * 




y + 7 _ Qy - 


- 10 



ALGEBRA. 135 

(255) Let x = the numerator of the fraction. 

y = the denominator of the fraction. 

Then, — = the fraction. 

y 

2x 2 
From the conditions, ■ — — r: = — , (1) 

y -\- 7 3 w 

and — — = -. (2) 

2j/ 5 w 

Clearing (1) and (2) of fractions, and transposing, 

Qx = 2y + 14, (3) 
and 5x=Qy — 10. (4) 

Solving for x, x — -+—>- — = — ^ — . (5) 



(6) 



Equating (5) and (6), 

Clearing of fractions, 5y -f- 35 = 18y — 30 

whence, 13y — 05, 

or, j/ = 5. 

Substituting tnis value of y in (3), 

6x= 10+14 = 24; 

whence, x = 4. 

4 
Therefore, the fraction is — . Ans. 

o 

(256) Let x = digit in tens place. 

j/ = digit in units place. 
Then 10^ + y = the number. 
From the conditions of the example, 

10;r -\- y = 4:(x-\-y) = 4=x -f 4j ; 
whence, 3y — G/r, 
or jk = %x. 
From the conditions of the example, 

10.r+j/ + 18= lOy +^r; 
whence, 9/ — 9.r = 18. 



136 ALGEBRA. 

Substituting the value of y, found above, 
18* — 9* = 18; 
whence, x = 2. 

jj/ = 2* = 4. 

Hence, the number = 10* + / = 20 + 4 = 24. Ans. 

(257) Let * = greater number. 

y = less number. 
Then, *+4 = 3ij, (1) 

and /+8 = jf. (2) 

Clearing of fractions, 4*+ 16 = 13/, 
and 2j/ + 16 = *; 
whence, 13y — 4* = 16. (3) 

2y— x = — 16. (4) 

Multiplying (4) by 4, and subtracting from (3) 

5y = 80, 
or y = 16. Ans. 
Substituting in (4), 32 — * = — 16; 

whence, * = 48. Ans. 



LOGARITHMS. 

(QUESTIONS 258-272.) 



(258) First raise ^ to t he . 29078 power. Since ^ = % 



/200V 29078 

Uooj 



2' 29078 , and log 2' 29078 = .29078 X log 2 = .29078 X 

.30103 — .08753. Number corresponding = 1.2233. Then, 

(200V 29078 
— j = 1 - 1.2233 = - .2233. 

We now find the product required by adding the log- 
arithms of 351.36, 100, 24, and .2233, paying no attention to 
the negative sign of .2233 until the product is found. (Art. 

647.) 

Log 351.36 = 2.54575 
log 100 = 2 
log 24=1.38021 

log .2233 = 1.34889 

j«» = 6.27485 = 

(/200\ • a9078 \ 
1 - ( Yqo ) ) 

Number corresponding = 188,300. 

The number is negative, since multiplying positive and 
negative signs gives negative; and the sign of .2233 is 
minus. Hence, 

x— — 188,300. Ans. 

(259) (a) Log 2,376 = 3.37585. Ans. (See Arts. 625 
and 627.) 

(b) Log .6413 =1.80706. Ans. 

(c) Log .0002507 = 4.39915. Ans. 

For notice of the copyright, see page immediately following the title page. 



138 LOGARITHMS. 

(260) (a) Apply rule, Art. 652. 
Log 755.4 = 2.87818 
log .00324 = 3. 51055 

difference = 5.36763 = logarithm of quotient. 
The mantissa is not found in the table. The next less 
mantissa is 36754. The difference between this and the 
next greater mantissa is 773 — 754=19, and the P. P. is 
763 — 754 = 9. Looking in the P. P. section for the column 
headed 19, we find opposite 9.5, 5, the fifth figure of the 
number. The fourth figure is 1, and the first three figures 
233; hence, the figures of the number are 23315. Since the 
characteristic is 5, 755.4 ~- .00324 = 233,150. Ans. 

(J?) Apply rule, Art. 652. 

Log .05555=^.74468 
log .0008601 = 4.93455 

difference = 1.81013 = logarithm of quotient. 
The number whose logarithm is 1.81013 equals 64.584. 
Hence, .05555 ^ .0008601 = 64.584. Ans. 
(c) Apply rule, Art. 652. 

Log 4.62 =_. 66464 
log. 6448= 1.80943 

difference = .85521 = logarithm of quotient. 
Number whose logarithm = .85521 = 7.1648. 
Hence, 4. 62 -4- . 6448 = 7. 1648. Ans. 

. 74 _ 238 X 1,000 

X ~ .0042- 6602 * 

Log 238 = 2.37658 

log 1,000 =_3. 

sum = 5.37658 = log (238 X 1.000) 
Log .0042 = 3.6 2325 
.6602 
124650 
373950 
373950 
.411469650 or .41147. 



(261) 



LOGARITHMS. 139 

.6602 
-3 



— 1.9 8 6 = characteristic. 
Adding, .41147 
-1.9806 



2.4308 7 (See Art. 659.) 
Then, log ( 238 00 X 42 1 ; 6 ^° ) = 5.37658 - 2.43087 = 6 94571 = 

6 94571 
74 log x\ whence, log x = — = 9.38609. Number 

whose logarithm = 9.38609 is 2,432,700,000 = x. Ans. 

(262) Log . 00743 = 3. 87099. 

log .006 =3.77815. 

4/. 00743 = log .00743 -*- 5 (Art. 662), and ^T006 = log 
.006 -^.6. Since these numbers are wholly decimal, we 
apply Art. 663. 

5 ) 3.8 7 09 9 

"1.5 7 41 9 = log \f. 00743. 

The characteristic 3 will not contain 5. We then add 2 
to it, making 5. 5 is contained in 5, 1 times. Hence, the 
characteristic is 1. Adding the same number, 2, to the 
mantissa, we have 2.87099. 2.87099 ^ 5 = .57419. Hence, 
log 4/. 00743 = 1.57419. 

.6) 3. 77815 .6 is contained in 3, — 5 times. 

5. .6 is contained in .77815, 1.29691 times. 

1.29691 



sum = 4.2 9 6 9 1 = log 4/. 006. 



Log 4/. 00743 = 1.57419 
log ^006 =4.29691 

difference = 3.27728 = log of quotient. 
Number corresponding = 1,893.6. 

Hence, ^.00743 ~ ^006 = 1,893.6. Ans. 
G. G. IV.—9 



140 LOGARITHMS. 

(263) Apply rule, Art. 647. 

Log 1,728 = 3.23754 

log .00024 = 4.38021 
log .7462 = 1.87286 
log 302.1 = 2.48015 
log 7.6094= .88135 



sum — 2.85211 = 
log (1,728 X . 00024 X . 7462 X 302. 1x7. 6094) . Number whose 
logarithm is 2.85211 = 711.40, the product. Ans. 



(264) Log 4/5.954= .77481 -^ 2 = .38741 

log 4^61.19 = 1.78668 -f3 = .59556 

sum = -98297 



log f 298. 54 = 2.47500 ~- 5 = .49500. 
4/5.954 X 4^ 61. 19 



Then, T ' y^^r — = lo g (V^Mi X /6L19) - log 

4/298.54 

^298.54 = .98297 - .49500 = .48797 = logarithm of the 
required result. 

Number corresponding = 3.0759. Ans. 



(265) 4/. 0532864 = log .0532864 -f- 7. 
Log .053_2864 = 2.72661. 

Adding 5 to characteristic 2 = 7. 

Adding 5 to mantissa = 5.72661. 

7-^-7 = 1. 

5.72661 -*-?= .81809, nearly. 

Hence, log {/0532864 = 1.81809. 

Number corresponding to log 1.81809 = .65780. Ans, 

(266) (a) 32 4 - 8 . 1.5 0515 
Log 32 = 1.50515. 4J3 

1204120 
602060 

7.224720 
7.22472 is the logarithm of the required power. (Art. 
657.) 

Number whose logarithm = 7.22472 is 16,777,000. 
Hence, 32 48 = 16,777,000. Ans. 



LOGARITHMS. 141 



(6) .76 s •".__ 

Log .76 = 1.88081. 1 + .880 81 

(See Arts. 658 and 659.) 3.6 2 

176162 
528486 
264243 



3.1885322 
3.62 



1.5 6 85 3 = log ,37028. 
Hence, . 76 3 - 6a = . 37028. Ans. 
(c) .84- 3 *. 

Log .84 = 1.92428. 1 + .9 2 4 2 8 

.38 



739424 

277284 

.3 5 122 64 

-.38 



1.9 7 12 3 = log. 93590. 
Hence, .84* 38 = .93590. Ans. 

6 /~l — 5 /23 

(267) Log y ■—— — log y — = logarithm of answer. 

/c49 71 

Log ^X = l-(logl-log 249)= ^(0-2.39620) = -. 39937 
= (adding + 1 and - 1)1.60063. 

Log y% = \ (log 23 - log 71) =1(1.36173 - 1.85126) = 

\ (-.48953) = - .097906 = (adding + 1 and - 1) l". 902094, 

or 1.90209 when using 5-place logarithms. 

Hence, 1. 60063-T. 90209 =T. 69854= log . 49950. Therefore, 

4^1=: .49950. AnS - 

(268) The mantissa is not found in the table. The next 
less mantissa is .81291; the difference between this and the 






142 LOGARITHMS. 

next greater mantissa is 298 — 291 = 7, and the P. P. is 
.81293 — .81291 = 2. Looking in the P. P. section for the 
column headed 7, we find opposite 2.1, 3, the fifth figure of 
the number; the fourth figure is 0, and the first three 
figures, 650. Hence, the number whose logarithm is. 81 293 
is 6.5003. Ans. 

2.52460 = logarithm of 334.65. Ans. (See Art. 640.) 
1.27631 = logarithm of .18893. Ans. We choose 3 for 
the fifth figure because, in the proportional parts column 
headed 23, 6.9 is nearer 8 than 9.2. 

(269) The most expeditious way of solving this ex- 
ample is the following: 

p v 1 -' 1 =p 1 v 1 1 - il , or v x ='\/l^l = v'^ L 

A A 



Substituting values given, v x = 1.495 



*-*V 134.7 
y 16.421 



t i i ,„* , log 134.7 -log 16.421 _,„. , 
Log v x = log 1.495 H 2 fTi = .17464 + 

2 - 1293y i ~ 1 1 - 21540 = . 17464 + . 64821 = . 82285 = log 6. 6504 ; 
whence, v x = 6.6504. Ans. 



e»n\\ t //7.1895 X 4,764.2 2 X 0.00326 5 1 n „ 1Q _ 
(270) Logf t()0Q489x 4573x ^ = -[lo g 7.189o 

+ 2 log 4,764.2 + 5 log .00326 — (log .000489 -f 3 log 457 + 2 

log . 576)] = K 77878 T 4 18991 = 2. 31777 = log . 020786. Ans 
5 

Log 7.1895= .85670 

2 log 4,764.2= 2 X 3.67799 = _7. 35598 

5 log .00326 = 5 X 3.51322 = 13.56610 

sum- 5.77878 

Log .000489=4.68931 
3 log 457 = 3 x 2.65992 = 7.97976 
% log .576 = 2x1. 76042 = 1. 52084 



sum = 4.18991. 



LOGARITHMS. 143 



(271) Substituting the values given, 
8,000 



X 



(ST »■«»(£)"' 



r l%$ x 2.25 2.25 

Log/ = log 8,000 + 2.18 log-| - log 2.25 = 3.90309 + 2.18 

(log 3 - log 16) - .35218 = 3.55091 + 2.18 X (.47712 =-- 
1.20412) = 1.96605 = log 92.480. Ans. 



(272) Solving for,, t = 'i^g^. 
Substituting values given, 



2.18/ 44 



1,000 

log. 044 2.64345 -2. 18 + .82345 

ooer t = — = = ■ 

8 2.18 2.18 2.18 

1.37773 = log .23863. Ans. 



Geometry and Trigonometry. 



(1) When one straight line meets another straight line 
at a point between the ends, the sum of the two adjacent 
angles equals two right angles. Therefore, since one of 
the angles equals -f of a right angle, the other angle 
equals two right angles (or ^-) minus -|. We have, then, 
} T °- — | = -§, or 1 J- right angles. Ans. 

(2) It is an isosceles triangle, since the sides opposite 
the equal angles are equal. 

(3) A regular decagon has 10 equal sides; therefore, the 
length of one side is 40 -~ 10 = 4 in. Ans. 

(4) The sum of all the interior angles of any polygon 
equals two right angles, multiplied by the number of sides 
in the polygon less two. As a regular dodecagon has 
12 equal sides, the sum of the interior angles equals two 
right angles X 10 ( — 12 — 2), or 20 right angles. Since 
there are 12 equal angles, the size of any one of them equals 
20 ~ 12, or If right angles. Ans. 

(5) Equilateral triangle. 

{iS) Since the two angles A and C, Fig. I, 
are equal, the triangle is isosceles, and a line 
drawn from the vertex B will bisect the line 
A C, the length of which is 7 inches; there- 
fore, 

AD = I) C=7 + 2= U in. Ans. 




(7) The length of the line = |/l2 9 - 9 a -f- /l5' J - 9\ or 
19.94 in. Ans. 

For notice of copyright, see page immediately following the title page. 



146 



GEOMETRY AND TRIGONOMETRY. 



(8) One of the angles of an equiangular octagon is equal 
to £ of 12 right angles, or 1^ right angles, since the sum 
of the interior angles of the equiangular octagon equals 
12 right angles. 

(9) See Art. 71. 

(10) In Fig. II, AB = 4: inches, an d O A = 6 in ches. 
We first find the length of O D. OD = VITA" - JTA*; but 

~OA* = 6 2 , or 36, and DA* = (f\\ or 4; 

therefore, O D = |/36 - 4, or 5.657. 

DC= O C- O D, or DC= 6 -5.657, 
or .343 inch. In the right triangle ABC, 
we have A C, which is the chord of half 
the arc A C B, equals y2 2 + .343 2 , or 
2.03 in. Ans. 

(11) Given, O C = 5f inches and 
OA = 17 > 2 = 8J inches, to find A B (see Fig. III). A C, 
which is one-half the chord A B, equals 
V~OA 2 - ~OV\ therefore, 
AC 




Fig. II. 



(5f) 2 , or 6.26 inches. 




Fig. III. 



Now, AB = 2 X AC; therefore, A B 

— 2 X 6.26, or 12.52 in. Ans. 

( 1 2) The arc intercepted equals f of 4, 
or 3 quadrants. As the inscribed angle 
is measured by one-half the intercepted arc, we have 3 -r- 2 
= \\ quadrants as the size of the angle. 

(13) Four right angles -h f = 4 X |, or 14 equal sectors. 

(14) Since 24 inches equals the perimeter, we have 
24 -r- 8 = 3 inches, as the length of each side or chord. 

Then, \/ (!) + ^'^ = 3 ' 92 in '' radius ' 

3.92 X 2 = 7.84 in., diameter. Ans. 

(15) In 19° 19' 19" there are 69,559 seconds, and in 
360°, or a circle, there are 1,296,000 seconds. Therefore, 



GEOMETRY AND TRIGONOMETRY. 147 

69 559 
69,559 seconds equal ' ' or .053672 part of a circle. 

' ' Ans. 

(16) Referring to Fig. 75 of the text and using the 
values given in the example, we have A B — 26 feet 7 inches, 
or 26.583 feet; A C = 40 feet; and the included angle 
A = 36° 20' 43". Then in the right triangle A D B, A B is 
known, and also the angle A. 

Hence, by rule 1, Arc. 98, B D = 26. 583 X sin 36° 20' 43" 
= 26.583 X .59265 = 15.754 feet. By rule 3, Art. 98, 
A D = 26.583 X cos 36° 20' 43" = 26.583 X .80546 = 21.411. 
A C— A D = 40 — 21.411 = 18.589 feet = D C. In the 
right triangle CDB, the two sides BD and DC are 

known; hence, tan C = 777^= c ' Q = .84749, and angle C 

U C lo. 589 

= 40° 16' 52". Ans. 

a 1 ■ ToA.r.oz?^ B D 15.754 

Applying rule 2, Art. 98, BC = -. — 7 ,= -. — — 

rr J sin 6 sin 40 16 52 

= ^77^7 = 24.37, or 24 ft. 4.4 in. Ans. 
.64654 

Angle B = 180° - (36° 20' 43" + 40° 16' 52") = 180° - 

76° 37' 36" = 103° 22' 25". Ans. 

(17) See Fig. 76 of the text. Solving the triangle ABC, 
we first find B D. By rule 1, Art. 98, BD = A B x sin A 
= 16fV X sin 54° 54' 54" = lQfy X .81830 = 13.434 feet. 

Sin BCD = ^ = jo' t!f = .99202 ; whence, angle BCD 

= 82° 45' 30". Ans. 

By rule 3, CD = B C X cos C— 13-? \ X cos 82° 45' 30" 
= 13J| X .12605 = 1.7069 feet. 

By rule 3, A D — A B x cos A = 16 T 5 2 X cos 54° 54' 54" 
= 16 T V X .57479 = 9.4361 feet. 

In the triangle ABC, the angle A C B is the supplement 
(see Art. 27) of the angle BCD and equals 180° — angle 
B CD, or A CB= 180° - 82° 45' 30" = 97° 14' 30". Ans. 

Angled BC= 180°- (angle BA C + angle A C B) = 180° 
- (54° 54' 54" + 97° 14' 30") = 180° - 152° 9' 24" = 27° 
50' 36". Ans. 






148 GEOMETRY AND TRIGONOMETRY. 

Side ,4 C=AD- CD, or A C= 9.4361 - 1.7069 = 7.7292 
= 7 ft. 8f in. Ans. 

For the triangle ABC, angle C ' = B C D (isosceles tri- 
angle), orC' = 82° 45' 30". 

Angle A B C = 180° - (angle A + angle C) = 180° 
- (54° 54' 54" + 82° 45' 30") = 180° - 137° 40' 24" 
= 42° 19' 36". Ans. 

AC = AD+C'D= 9.4361 + 1.7069 = 11.143 = 11 ft. 
If in. Ans. 



(18) If one-third of a certain angle equals 14° 47' 10", 
then the angle must be 3 X 14° 47' 10", or 44° 21' 30". 
%l X 44° 21' 30", or 110° 53' 45", equals one of the other 
two angles. The third angle equals 180° — (110° 53' 45" 
+ 44° 21' 30"), or 24° 44' 45". 

(19) Referring to Fig. 70 of the text, let B C = 437 feet 
and A C = 792 feet, to find the hypotenuse A B and the 
angles A and B. 

AB= VA~C 2 + ~BV = 4/792 2 + 437 2 = 4/818,233 = 904 ft. 
6f in. Ans. 

_ a ^^ a side opposite . 437 

By rule 4, Art. 98, tan A = -^ ££ , or tan A = — 

J side adjacent 792 

= .55177; therefore, A = 28° 53' 19". Ans. 

Angle B = 90° - 28° 53' 19", or 61° 6' 41". Ans. 

(20) See Fig. IV. Angle B = 180°- (29°21'+76° 44' 18") 

= 180°-106°5'18" = 73°54'42". 

From C, draw C D perpen- 
\ dicular to A B. 
\T% A D = A C cos A = 31.833 
9 $M X cos 29° 21'= 31.833 X .87164 

§. I >N U =27. 747 feet. CD = ACs'mA 

^ = 31.833 X sin 29° 21' = 31.833 
X .49014= 15.603. 

l*l 6 ™ t lf , = 16.24 feet = 16 ft. 3 in. 
sin 73° 54' 42" 

15.603 

= 4.5 feet. 




tan B tan 73° 54' 42" 



GEOMETRY AND TRIGONOMETRY. 



149 



A B = 
3 inches. 



(21) 
d = 



A D + D B = 27.747 + 4.5 



Ans. 



32.247 = 32 feet, 
BC= 16 ft. 3 in. 
AB-32 ft. 3 in. 

^ ^ 73° 54' 42" 



By rule, Art. 135, 



A 



89.42 



\' 113. 8528, or 10.67 in. Ans. 



7854 V -7854 
Circumference equals 10.67 X 3.1416, or 33.52 in. Ans. 
In a regular hexagon inscribed in a circle, each side is 

equal to the radius of the circle; therefore, — '- — — 5. 335 in. 
is the length of a side. Ans. 



T Vof 360 c 



or 221°. m O = \ of 




(22) Angle m O B 
m n — \oi 2, or 1 inch. See Fig. V. 

Side m B — O m X tan 22£°, or m B 
= 1 X .41421 = .41421. 
• A£ = %mB; therefore, A B= . 82842 
inch. 

Area of A O B = \ x .82842 X 1 = 
.41421 square inch, which, multiplied by 
8, the number of equal triangles, equals 
3.31368 square inches. 

Volume of bar = 3.31368 X 10 X 12 = 397.6416 cu. in. 

Weight of bar equals 397.6416 X .282 = 112.1349, or 
112 lb. 2 oz. Ans. 

(23) .5236 X 16 3 = 2,144.66 cubic inches equals the vol- 
ume of a sphere 16 inches in diameter. 

.5236 X 12 3 = 904.78 cubic inches equals the volume of a 
sphere 12 inches in diameter. 

The difference of the two volumes equals the volume of 
the spherical shell, and this multiplied by the weight per 
cubic inch equals the weight of the shell. Hence, we have 
(2,144.66 - 904.78) X .261 = 323.61 lb. Ans. 

(24) The circumference of the circle equals ? ^ , 

or 72.0833 inches. The diameter, therefore, is ' — , 

or 22.95 in., nearly. Ans. 



150 GEOMETRY AND TRIGONOMETRY. 

(25) (a) lT^x' inches = 17. 01G inches. 

Area of circle = .7854^ a = .7854 X 17.016 2 = 227.41 sq. in. 

Ans. 
(«5) 16° T 21" = 16.1225°. By rule, Art. 132, 

. -dn 3.1416X17.016X16.1225 „ on , . A 

/= — — = — — = 2.394 m. Ans. 

360 360 

(26) (a) By rule. Art. 144, area = 12 X 8 X .7854 
= 75.4 sq. in. Ans. 

(/;) Applying the formula, Art. 1 43, a = 6, £.= 4, D 

- 6 ~ 4 - 1 - 1 
~~ 6 + 4 — 10 — 5' 

Perimeter = 3.1416 (6 + 4) ^-lM* ? = 31. 731 in. Ans. 

(27) Area of base = .7854 X 7 2 = 38.484 square inches. 
Slant height of cone = 4/H 2 -J- 3.V\ or 11.5434 inches. 
Circumference of base = 7 X 3.1416 = 21.9912 inches. 

Convex area of cone = 21.9912 X — — ; = 126.927 square 

2 

inches. 

Total area = 126.927 + 38.484 = 165.41 sq. in. Ans. 

(28) Volume of sphere equals .5236 X 10 3 = 523.6 cubic 
inches. 

Area of base of cone = .7854 X 10 a = 78.54 square inches. 
3 X 523.6 



78. 54 



= 20 in., altitude of cone. Ans. 



(29) Volume of sphere = .5236 X 12 3 = 904.7808 cubic 
inches. 

Area of base of cylinder = .7854 X 12 2 = 113.0976 square 

inches. 

tt • u. r 1- A 904.7808 D . . 

Height or cylinder = = 8 in. Ans. 

a r* v p n 

(30) {a) Area of the triangle equals , or 

H x 12 „_ . . 

-=— = o7 sq. in. Ans. 



GEOMETRY AND TRIGONOMETRY. 



151 




|/12 2 + 7.247 2 = 4/196.519 



(b) See Fig. VI. Angle BAD = 79° 22'; angle A BD 
= 90° - 79° 22' = 10° 38'. Side A B b 

= i? £> -f- sin 79° 22' = 12 ~ .98283 = 
12.209 inches. 

Side A D = B D x tan 10° 38' rr 12 
X .18775 = 2.253 inches. 

Side DC=AC-AD=9.5- 2.253 A 
= 7.2-47 inches. 

Side B C = VUB* -f UC 
= 14.018 inches. 

Perimeter of triangle equals A B + B C+ A C = 12.209 
+ 14.018 + 9.5 = 35.73 in. Ans. 

(31) The diagonal divides the trapezium into two tri- 
angles; the sum of the areas of these two triangles equals 
the area of the trapezium, which is, therefore, 

11 X 7 . 11 X 4J _ 17 . A 

— ^ 1 2~"^ = 61J sq. in. Ans. 

(32) Referring to Fig. Ill, example 11, we have O A or 
<9Z?=10-^2=5 inches, and A B = 6f inches. 

C B 6f -J-" 2 



Sin C<9^ 



67500; therefore, angle C6>i? 



6>£ 5 

= 42° 27' 14.3". 

Angle A OB= (42° 27' 14. 3") X 2 = 84° 54' 28. 6". Ans. 
<9C=6>£xcosC<9£ = 5x .73782 = 3.6891. 

Area of sector = 10 2 X . 7854 X 84 ot/f' 6 = 18. 524 square 
. obO 

inches. 



Area of triangle = 



6.75 X 3.6891 

2 



12.450 square inches. 



18.524 - 12.450 = 6.074 sq. in., the area of the seg- 
ment. Ans. 

(33) See Fig. VII. Area of lower base 
= 18 2 X .7854 = 254.4696 square inches. 

Area of upper base=12" X .7854=113.0976 
square inches. 

EG= BG- A F=9 - 6, or 3 inches. 

Slant height FG = ViTG' + JTP = 




Fig. VII. 



4/3 2 + 14" = 14.32 inches. 



152 



GEOMETRY AND TRIGONOMETRY. 



By formula, Art. 169, 
r _(p +p')s _ 37.6992 + 56.5488 



X 14.32 = 674.8156 sq. in. 



2 2 

Total area = 674.8157 + 254.4696 + 113.0976 = 1,042.38 
sq. in. Ans. 

By formula, Art. 1 70, V= {A 



h 



■\-tfA X a)- = (113.0976 
o 

+ 254.4696 + ^113.0976 X 254.4696) - 1 / = 2,506.997 cu. in. 

Ans. 

(34) Area of surface of sphere 27 inches in diameter 
= 7T d 2 = 3.1416 X 27 2 = 2,290.2 sq. in. Ans. 

(35) Area of end = 19 2 X .7854 = 283.5294 square 
inches. Volume = 283.5294 X 24 = 6,804.7056 cu. in. 

Ans. 




Fig. VIII. 



(36) Given, IB = 2 inches and HI 
= I K = ^- = 7 inches to find the radius. 
See Fig. VIII. 
IB : HI = HI: AI, or 2 : 7 = 7 : A I. 

Therefore, A I — - 4 / = 24£ inches. 
AB = A I+IB= 24^-f 2 = 26^ inches. 
_ A B _ 26j 

"~2~ ~ 2 



Radi 



13J- in. Ans. 



(37) (a) In Fig. IX, given O B = ™ or 8 inches, and O A 
= 13 -f- 2 = 6| inches, to find the volume, area, and weight. 

Radius of center circle equals , 

2 

or 7| inches. Length of center line 
= 3.1416 x 2 X 7J= 45. 5532 inches. The 
radius of the inner circle is 6^ inches, 
and of the outer circle 8 inches; there- 
fore, the diameter of the cross-section 
on the line A B is 1\ inches. Then, 
the area of the imaginary cross-section is {1\Y X .7854 
= 1.76715 sq. in. 

Volume of ring = 1.76715 X 45.5532 = 80.499 cu. in. Ans. 

(b) Weight of ring = 80.4993 X .261 = 21 lb. Ans. 




Fig. IX. 



GEOMETRY AND TRIGONOMETRY. 



153 



(38) The convex area = 4 X 5 J- X 18 = 378 sq. in. Ans. 
Area of the bases = 5^ X 5^- X 2 = 55.125 square inches. 
Total area = 378 + 55.125 = 433.125 sq. in. Ans. 
Volume = (5i) 2 X 18 = 496.125 cu. in. Ans. 



(39) In Fig. X, O C = 



A C 



tan 30° 

since A O C = \ of A OB, A O C '= 30°.) 
6 



(| of 360° = 60°, and 



OC = 



57735 



= 10.392. 



Area of A OB = 



12 X 10.392 

2 




Fig. X. 



■ = 62.352 square feet. 
Since there are 6 equal triangles in a 
hexagon, then the area of the base 
= 6 X 62.352, or 374.112 square feet. 
Perimeter = 6 X 12, or 72 feet. 

Convex area = ■ = 1,332 sq. ft. Ans. 

Total area = 1,332 + 374.112 = 1,706.112 sq. ft. Ans. 



(40) Area of the base— 374.112 square feet, and alti- 
tude = 37 feet. Since the volume equals the area of the 
base multiplied by -J- of the altitude, we have 



37 
volume = 374.112 X — = 4,614 cu. ft. 
o 



Ans. 



(41) Given, A B = 6| inches, and O B = O A = 10 +- 2 
= 5 inches (see Fig. XI), to find the area of the sector. 
Area of circle = 10 2 X .7854 = 78.54 square inches. 




Sin A OC = 



AC 6£-*- 2 



.68750 



Fig. XI. 



OA ~ 5 
therefore, A O C= 43° 25' 57". 

A OB=2 X A OC=2X 43° 25' 57" 
= 86° 51' 54" = 86.865°. 

By rule, Art. 137, 

86.865 wo ^ 

-TT777T- X 78.54 = 18.95 sq. in. Ans. 
ooO 



154 GEOMETRY AND TRIGONOMETRY. 

(42) By rule, Art. 1 29, A = bh — 7 X 10f (129 inches 
= 10| f eet) = 75^ sq. ft. Ans. 

(43) See Art. 130. 

Area of trapezoid = ({—^) A = 15 tV + 21H x 7J = U3 ?5 

sq. ft. Ans. 

(44) (a) Side of square having an equivalent area 
= i/143.75 = 11.99 ft. Ans. 

(#) Diameter of circle having an equivalent area 
143.75 



7854 



4/183.0277= 13J ft. Ans. 



■(*) Perimeter of square = 4 X 11.99 = 47.96 ft. 
Circumference of circle = 13 hX 3.1416 = 42.41 ft. 



Difference of perimeter = 5.55 ft. =5 ft. 6. 6. in. 

Ans. 

B 

(45) In the triangle ABC, 
Fig. XII, AB = 24, feet, B C 
= 11.25 feet, and A C= 18 feet. 
m -\- 11 : a -\- b = a — b : in — n 
or 24 : 29.25 = 6.75 : ;« - n. 



29.25X6.75 - 

7/j — n = — = 8.226562. 

/o4 

Adding ;;/ -(- w and ;// — a, we have 

02 -f n = 24 

m — n — 8.226562 




2 m = 32.226562 
m = 16.113281. 



Subtracting w — n from 7;/ -}- n, we have 

2/; = 15.773438 
«= 7.886719. 



GEOMETRY AND TRIGONOMETRY. 155 

In the triangle ADC, side A C = 18 feet, side A D 
= 16.113281; hence, according to rule 2, Art. 98, cos A 

= 1 ?^ 1 |^ 1 = .89518, or angle A = 26° 28' 5". In the tri- 
lb 

angle BDC, side BD = 7.886719, and side B C= 11.25 feet. 

7 88671 9 
Hence, cos B = -^t^~ = .70104, or angle B = 45° 29' 23". 
11. Zo 

Angle C= 180° - (45° 29' 23" + 26° 28' 5") = 108° 2' 32". 

(A — 26° 28' 5". 

Ans. \ B = 45° 29' 23". 

( C = 108° 2' 32". 



IV.— 10 



NOTICE 

The present set of answers on the subject of Geometry 
and Trigonometry are less in number than were contained in 
the former edition. As a consequence, there is a slight break 
in the page numbers between the last page of the answers on 
Geometry and Trigonometry and the Paper following. 



ELEMENTARY MECHANICS. 

(QUESTIONS 355-453.) 



(355) Use formulas 18 and 8. 
Time it would take the ball to fall to the ground = t = 

S* ./2X 5.5 KQAQ . 
— = y — - — — - = .58484 sec. 
^" o2.1o 

The space passed through by a body having a velocity of 

500 ft. per sec. in .58484 of a second = 5 = F/=500X 

.58484 = 292.42 ft. Ans. 

(356) Use formula 7. 

— '— — = 55. 85 ft. per sec. Ans. 

60 

8 1 

(357) 160 -f- 60 X 7 = k5T revolution in - sec. 360° X 

21 7 

l = 13?i-°= 137° 8' 34§-'. Ans. 
/cl ( 7 

(358) (a) See Fig. 20. 36" = 3'. 4 -^ 3 = i = number 

of revolutions of pulley to one revolution of fly-wheel. 

4 
54 X o = 72 revolutions of pulley and drum per min. 100 — 

o 

I — X 3.1416 1 = 21.22 revolutions of drum to raise elevator 

21 22 
100 ft. -—— X 60 = 17.68 sec. to travel 100 ft. Ans. 
7 <> 

(b) 21.22 : x :: 30 : 60, or x = 21 - 2 * X 6 ° = 42.44 rev. 

0\) 

For notice of the copyright, see page immediately following the title page. 



166 ELEMENTARY MECHANICS. 

per min. of drum. The diameter of the pulley divided by 




Fig. 20. 



the diameter of the fly-wheel = -, which multiplied by 



42.44= 31.83 revolutions per min. of fly-wheel. 
(359) See Arts. 857 and 859. 
See Art. 861. 
See Arts. 843 and 871. 
See Art. 871. 



Ans. 



(360) 

(361) 
(362) 
(363) 



See Arts. 842, 886, 887, etc. 

The relative weight of a body is found by comparing it 
with a given standard by means of the balance. The abso- 
lute weight is found by noting the pull which the body will 
exert on a spring balance. 

The absolute weight increases and decreases according to 
the laws of weight given in Art. 890 ; the relative weight is 
always the same. 

(364) See Art. 861. 

(365) See Art. 857. 



ELEMENTARY MECHANICS. 167 

(366) See Art. 857. 

(367) If the mountain is at the same height above, and 
the valley at the same depth below sea-level respectively, 
it will weigh more at the bottom of the valley. 

(368) ^~ = 6 miles. Using formula 12, d 2 : R* :: 

jjz u R " W 3,960 2 X 20,000 . ' 

W\ w,weh3.vew = —j r - = — ' = 19,939.53 + lb. 

= 19,939 lb. 8-oz. Ans. 

Z 

(369) Using formula 11, R : d :: W : w y we have 
w = rf^ = 3,958x80,000 = 1 m89 lb = , b u _l 

K o,9b0 4 
oz. Ans. 

(370) See Art. 870. 

(371) See Art. 894. 

(372) The velocity which a body may have at the in- 
stant the time begins to be reckoned. 

(373) Because the man after jumping tends to continue 
in motion with the same velocity as the train, and the sud- 
den stoppage by the earth causes a shock, the severity of 
which varies with the velocity of the train. 

(374) See Arts. 870 and 871. 

(375) See Art. 872. 

(376) That force which will produce the same final 
effect upon a body as all the other forces acting together is 
called the resultant. 



(377) (a) If 


a 5-in. 


line = 


: 20 lb., a 


l-in. 


line = 


4 1b, 


l-r4 = j in. = 
4 


= 1 lb. 


Ans. 


(*) 4- 


f- 4 = 


= 1.562£ 


I in. 


6^ lb. Ans. 
4 















(378) Those forces by which a given force may be 



168 



ELEMENTARY MECHANICS. 



replaced, and which will produce the same effect upon a 
body. 

(379) Southeast, in the direction of the diagonal of a 
East square. See Fig. 21. 

(380) 4' 6' = 54". 54x2x|x 

.261 = 21.141 lb. = weight of lever. 
Center of gravity of lever is in the 
middle, at a, Fig. 22, 27" from each 
end. Consider that the lever has no 
weight. The center of gravity of 
the two weights is at b f at a dis- 

47 v 54 
tance from c equal to ' ^ = 21.508" = bc. Formula 20, 

Art. 911. 




Fig. 21. 



-54T 



|< 1 27 >± h 



■J/83. 



W; 



w \ 



b\ 



-21.508—- 




71lb. 



Pig. 22. 

Consider both weights as concentrated at b, that is, 

imagine both weights removed and replaced by the dotted 

weight W, equal to 71 + 47 = 118 lb. Consider the weight 

of the bar as concentrated at a, that is, as if replaced by 

a weight w — 21.141 lb. Then, the distance of the balancing 

21.141 X 5.492 OOA „ . 
point /, from e, or fe, = 2L141 + n8 = -834 , since ae = 

27-21.508 = 5.492". Finally, /*+*//=/// = . 834 + 21.508 
= 22.342" = the short arm. Ans. 54-22.342 = 31.658" = 
long arm. Ans. 



ELEMENTARY MECHANICS. 
(381) See Fig. 23 



169 




75 lb. 



Fig. 23. 



(382) See Fig. 24. 




(383) 46 — 27 = 19 lb., acting in the direction of the 
force of 46 lb. Ans. 



170 



ELEMENTARY MECHANICS. 



(384) (a) 18 X 60 X 60 = 64,800 miles per hour Ans 
\b) 64,800 X 24 = 1,555,200 miles. Ans. 




Fig. ^5. 



ELEMENTARY MECHANICS. 



171 



(385) (a) 15 miles per hour = 6Q x ^ = 22 ft. per 

sec. As the other body is moving 11 ft. per sec., the 
distance between the two bodies in one second will be 
22 -j- 11 = 33 ft., and in 8 minutes the distance between 
them will be 33 X GO X 8 = 15,840 ft., which, divided by the 

number of feet in one mile, gives ' = 3 miles. Ans. 

5, 280 

(b) As the distance between the two bodies increases 

33 ft. per sec, then, 825 divided by 33 must be the time 

825 
required for the bodies to be 825 ft. apart, or 

sec. Ans. 



33- = 35 



(386) See Fig. 25. 

(387) (a) Although not so stated, the velocity is 
evidently considered with reference to a point on the shore. 
10 — 4 = 6 miles an hour. Ans. 

(b) 10 + 4 = 14 miles an hour. Ans. 

(c) 10 — 4 + 3 = 9, and 10 + 4+ 3 = 17 miles an hour. Ans. 

(388) See Fig. 26. 




Fig. 26. 



172 



ELEMENTARY MECHANICS. 



(389) See Fig. 27. By rules 2 and 4, Art. 754, be = 
87 sin 23° = 87 X .39073 = 33.994 lb., a c = 87 cos 23° = 
87 X .92050 = 80.084 lb. 




80 lb. 



Fig. 27. 



(390) See Fig. 28. (b) By rules 2 and 4, Art. 754, 
* = 325 sin 15° = 325 X .25882 = 84.12 lb. Ans. 

(a) a ^- = 325 cos 15°'= 325 X. 96593 = 313.93 lb. Ans. 




Fig. 28. 



(391) Use formula lO. 
W 125 



m = 



g 32.16 



= 3.8868. Ans. 



ELEMENTARY MECHANICS. 173 

IV 

(392) Using formula lO, m = — , W = mg=z 53.7 X 

32.16 = 1,727 lb. Ans. 

(393) (a) Yes. (&) 25. (c) 25. Ans. 

(394) (a) Using formula 12, d* : R* :: IV : W, d = 

^WW = ^ 4,000-X 141 = 9 m . les _ ^ m __ 

w 100 

4,000 = 749.736 miles. Ans. 

(b) Using formula 11, R : d :: W : w, d = —rjr — 
4,000 X 100 



141 

miles. Ans. 



= 2,836.88 miles. 4,000 - 2,836.88 = 1,163.12 



(395) (a) Use formula 18, 

A /%h J% X 5,280 ^ „ ' . 

t = y ■ — = y — ittttt- = 18.12 sec. Ans. 
£■ 32.16 

(£) Use formula 1 3, v = gt = 32. 16 X 18. 12 = 582. 74 ft. per 
sec., or, by formula 16, v = i/2gk = */% X 32.16 X 5,280 = 
582.76 ft. per sec. Ans. 

The slight difference in the two velocities is caused by not 
calculating the time to a sufficient number of decimal places, 
the actual value for / being 18.12065 sec. 

(396) Use formula 25. Kinetic energy = Wh = -= — . 

Wh = 160 X 5,280 = 844,800 ft. -lb. 

Wv> 160X582.76 2 ,,,-„,,„ . 
-TF= 2X32.16 -844,799 ft. -lb. Ans. 

(397) (a) Using formulas 15 and 14, 

h = fe = a x 38 16 = 86,593 ft - = 16 ' 4 miles - Ans - {b) 

V 

t——= time required to go up or fall back. Hence, total 

o 

time = — sec. = — ?^-^ = 2.4461 mm. = 2 mm. 26.77 

£" 60 X 32.16 

sec. Ans. 









174 ELEMENTARY MECHANICS. 

(398) 1 hour = GO min., 1 day = 24 hours; hence, 1 

day = 60 X 24 = 1,440 min. Using formula 7, V ' — -; 

. T/ 8,000 X 3.1416 ... . 

whence, K = tttt, = 17.453 + miles per mm. Ans. 

1,440 r 

(399) (a) Use formula 25. 

ir- *■ ^' 400X1 ,875 X 1,875 Q1 ... _ QQ _ 
Kinetic energy =— -— = ■ = 21,863,339. 55 

2^* A X O/V. lb 

ft. -lb. Ans. 

,,, 21,863,339.55 _ AO , „_, . . 

(£) ' =10,931.67 ft. -tons. Ans. 

/v, uuu 

(<:) See Art. 961. 

Striking force X ^ = 21,863,339.55 ft. -lb., 

1/0 

or striking force = 21 » 863 * 339 - 55 = 43> 72 6, 679 lb. Ans. 



(400) Using formula 18, t = i/EA = i/I 



2 X 200 _ 



g r 32.16 
3.52673 sec, when^-= 32.16. 

t _ i/ 2x 200 _ 4 472U sec> when = 2Q> 
V 20 

4.47214 — 3.52673 = 0.94541 sec Ans, 

(401) See Art. 910. 

(402) See Art. 963. 

(403) (a) See Art. 962. 

~ *i H^ 800 „ _ W 500 

£>=77 = —- v= T1 f KE . Hence, ,Z? = — = 00 lflw 8oo 



F £^ 1,728 ' £•*/ 32.16x T 8 T °i> Q g 

33.582. Ans. (£) In Art. 962, the density of water was 

found to be 1.941. (c) In Art. 963, it is stated that the 

specific gravity of a body is the ratio of its density to the 

33 582 
density of water. Hence, -t-^-tt- == 17.3 = specific gravity. 

J. . y 4r J. 

If the weight of water be taken as 62.5 lb. per cu. ft., the 
specific gravity will be found to be 17.28. Ans. 

(404) Assuming that it started from a state of rest, 
formula 1 3 gives v = g t = 32. 16 X 5 = 160, 8 ft. per sec. 



ELEMENTARY MECHANICS. 175 

(405) Use formulas 17 and 13. h = ~ g f = 

2 

— — X 3 2 = 144.72 ft., distance fallen at the end of third 
2 

second. 

z; =£•/ = 32.16 X 3 = 96.48 ft. per sec, velocity at end of 
third second. 

96.48 X 6 = 578.88 ft., distance fallen during the remain- 
ing 6 seconds. 

144.72 + 578.88 = 723.6 ft. = total distance. Ans. 

(406) See Art. 961. 

Striking force X ^ = 8 X 8 = 64. Therefore, striking 
12 

CK 1 

force = — = 1,536 tons. Ans. 
12 

(407) See Arts. 901 and 902. 

(408) Use formula 19. 

Centrifugal force = tension of string = .00034 WRN* = 

1 K 

.00034 X (.5236 X 4 3 X .261) X ^ X 60 3 = 13.38+ lb. Ans. 

12 

(409) (SO 2 - 7G 2 ) X .7854 X 26 X .261 -i- 2 = weight of 
\ of rim. 

D 80-10 35 ^ 
R = Y^12 = 12 ft ' 

According to Art. 904, F= .00034 IVRN' + 3.1416 = 

.00034 X (80* - TO') *. 7854 X 26 X .261 x g x IW + 

3.1410 = 38,641 lb. Ans. 

(41 0) (a) Use formulas 11 and 12. R I dr. IV: w s 

ur wR 1x4,000 ,_„ . 
or W— —j- - = — —± — = 40 lb. Ans. 
a 100 

(b) d* : R* :: W : w, or w = ^^ ~ = 38.072 lb. Ans 



(411) See Art. 955. 
10,746 X 354 



10 X 33,00(? 



= 11.5275 H. P. Ans. 



176 ELEMENTARY MECHANICS. 

(412) Use formula 12. 

d* : R* :: W: w, or d= j A 00 ^ x 2 - 13)0 64 mi., nearly. 

1 6 

13,064 - 4,000 = 9,064 miles. Ans. 

(413) Use formula 18. / = /^ = |/%£# => 
lc 7634 sec., nearly. 

1.7634 X 140 = 246.876 ft. Ans. 

(414) oTy = q sec - Use formula 17. 



£ = !^ a = ! X 32.16 X ^V = 1.78| ft. = 1 ft. 9.44 in. 



Ans. 




(415) See Arts. 906, 907. 

(416) See Arts. 908, 909. 



ELEMENTARY MECHANICS. 



Ill 



(417) No. It can only be counteracted by another 
equal couple which tends to revolve the body in an opposite 
direction. 

(418) See Art. 914. 

(419) Draw the quadrilateral as shown in Fig. 29. 
Divide it into two triangles by the diagonal B D. The 
center of gravity of the triangle B C D is found to be at a, 
and the center of gravity of the triangle A B D is found to 
be at b (Art. 914). Join a and b by the line a b, which, on 
being measured, is found to have a length of 4.27 inches. 
From C and A drop the perpendiculars C F and A G on the 

diagonal B D. Then, area of the triangle A B D — - (AG X 

B D) } and area of the triangle B C D= \ (C Fx B D). 

Measuring these distances, B D=\\", C F=5.1\ and 

a G=i.r. 

Area.ABD=l X 7.7 X 11 = 42.35 sq. in. 



Area BCD= - X 5.1 X 11 = 28.05 sq. in. 

According to formula 20, the distance of O t the center of 

28.05 X 4.27 



gravity, from b is 



1.7. Therefore, the cen< 



28.05 + 42.35 
ter of gravity is on the line a b at a distance of 1.7' from b. 



(420) See Fig. 30. The 
center of gravity lies at the 
geometrical center of the penta- 
gon, which may be found as 
follows: From any vertex draw 
a line to the middle point of the 
opposite side. Repeat the 
operation for any other vertex, 
and the intersection of the two 
lines will be the desired center 
of gravity. 




Fig. SQ. 



178 



ELEMENTARY MECHANICS. 



(421) See Fig. 31. Since any number of quadrilaterals 
can be drawn with the sides given, any number of answers 
can be obtained. 

Draw a quadrilateral, the lengths of whose sides are equal 
to the distances between the weights, and locate a weight on 
each corner. Apply formula 20 to find the distance C 1 W 1 ; 

Measure the distance C x W^\ 



thus, C x W x = ^4? = 



9 + 21 




Fig. 31. 
suppose it equals say 36". Apply the formula again. 

12". Measure C a W % \ it equals say 
17 X 31.7 



C%C% 1 



3i. r. 

Apply the formula again 



C C — 

17 + 15 + 9 + 21 



8.7' 



C is center of gravity of the combination. 



(422) Let A B C D E, Fig. 32, be the outline, the 
right-angled triangle cut-off being E S D. Divide the 
figure into two parts by the line ;;/ u, which is so drawn 
that it cuts off an isosceles right-angled triangle m B n, 
equal in area to E S D, from the opposite corner of the 
square- 



ELEMENTARY MECHANICS. 



179 



The center of gravity of A mnC D E is then at C x , its 
geometrical center. Bm = 4 in. ; angle Bm r = 45° ; there- 
at m B 




fore, Br = Bin X sin Btnr = ± X .707 = 2.828 in. C 2i the 

2 

center of gravity of Bmn, lies on Br, and B C^ = —B r 

o 

-|x 2.828 = 1.885 in. B C x = A B X sin ^^^=14x 

sin 45° = 14 X .707 = 9.898 in. C, C, = BC X - B C^ = 9.898 
-1.885 = 8.013 in. 

Area 
4X4 



Area ABCDE= i4 3 - 1^-1=188 sq. in. Area mBn 



2 



sq. in. Area A mn CB E =l^S 



180 



2 
sq. in. 

The center of gravity of the combined area lies at C. at 
ft. G. IV.— 11 



180 



ELEMENTARY MECHANICS. 



a distance from C l , according to formula 20 (Art. 911) y 



equal to 



8 X C, C n 8 X 8.013 



180 + 8 
BC=BC-CC 



188 



= .341 in. CX- .341 in. 



9.898 - .341 = 9.557 inches. Ans. 



(423) {b) In one revolution the power will have moved 

through a distance of 2 X 15 X 3.1416 = 94.248", and the 

1" 
weight will have been lifted -r. The velocity ratio is then 



94.248-^ 
4 



376.992. 

376.992 X 25 = 9,424.8 lb. Ans. 
{a) 9,424.8 - 5,000 = 4,424.8 lb. Ans. 
(c) 4, 424. 8 -f- 9, 424. 8 = 46.95$. Ans. 

(424) See Arts. 920 and 922. 

(425) Construct the prism ABED, Fig. 33. From 
E, draw the line E F. Find the center of gravity of the 

A/ 




Horizontal 

Pig. 33. 



rectangle, which is at C v and that of the triangle, which is 



ELEMENTARY MECHANICS. 181 

at C~. Connect these centers of gravity by the straight line 
C l C 2 and find the common center of gravity of the body by 
the rule to be at C, Having found this center, draw the 
line of direction C G. If this line falls within the base, the 
body will stand, and if it falls without, it will fall. 

(426) (a) 5 ft. 6 in. = 66". 66 -r- 6 = 11 = velocity 
ratio. Ans. 

(b) 5 X 11 = 55 lb. Ans. 

(427) 55 X .65 = 35.75 lb. Ans. 

35 X 60 



(428) Apply formula 20. 5 ft. = 60". 



180 + 35 
9.7674 in., nearly, = distance from the large weight. Ans. 

(429) (a) 1,000 -f- 50 = 20, velocity ratio. Ans. See 
Art. 945. (b) 10 fixed and 10 movable. Ans. (c) 50 + 95 
= 52.63$. Ans. 

(430) PX circumference = Wx 5-, or 60 X 40 X 3.1416 

o 

= W X 4 or W= 60 X 40 X 3. 1416 X 8 = 60, 318. 72 lb. Since 

o 

the efficiency of combination is 40$, the tension on the stud 
would be .40 X 60,318.72 = 24,127.488 lb. Ans. 



(431) (a) ^20 2 +5 2 = 20.616 ft. = length of inclined 
plane. 

PX length of plane = W X height, or P X 20.616 = 
1,580 X 5. 

1 KQ() v % 

^= oncifi = 383 ' 2 1K AnS - ( /7 ) In the seCOnd Case » 
Z\j. bio 

P X length of base =Wx height, or Px 20 = 1,580 X 5; 
hence, P= 1,58 ° X 5 = 395 lb. Ans. 

(432) Wx 2 = 42 x 6, or W= 42 Q X 6 = 126 lb. 

126 + 42 = 168 lb. 168 X 1 = W X 12, or W = ^ = 14 lb. 

Ans. 



182 



ELEMENTARY MECHANICS. 



(433) See Fig. 34. Px 14 X 21 X 19 = 4 X 3^ X &| 

/w 4 o 



X 725, or 



p _ 2| X 3j X 2j- X 72 5 
P ~ 14X21X19 - = 3 - 03 ^ lb - Ans « 



TF 



: £4 J 



— fB^ 



6^. 182—^6'- 



-30*- 



P 

Fig. 34. 



J 



-iS-*-H 



o 



(434) See Fig. 35. (a) 35 X 15 X 12 X 20 = 5 X 3 i X 

3X^or 

W7 35 X 15 X 12 X 20 _ . __ „ . 
W = 5 X H X 3 = 2 ' 40 ° 1K AnS ' 




ELEMENTARY MECHANICS. 



183 



(b) 2,400 -r- 35 = G8y = velocity ratio. Ans. 

(c) 1,932 -^ 2,400 = .805 = 80.5^. Ans. 



(435) In Fig. 36, let the 12-lb. weight be placed at A 
the 18-lb. weight at B, and the 15-lb. weight at D. 
Use formula 20. 
12 X 15 



18 + 12 



6" = distance C X B = distance of center of 




Fig. 36. 



gravity of the 12 and 18-lb. weights from B. Drawing C\ V, 

C 1 D. Measuring the distances 



CC 



IbxC^D 



(12 +18) + 15 3 
of C from BI), DA, and A B y it is found that Ca = 3.45", 
Cb = 5.25", and Cd= 4.4". Ans. 



(436) (tf) Potential energy equals the work which the 
body would do in falling to the ground = 500 X 75 = 
37,500 ft. -lb. Ans. 



184: 



ELEMENTARY MECHANICS. 

T/i _ J"l X 75 



(b) Using formula 18, / = 



g 



32.16 



= 2.1597 



sec. = -035995 min. 5 the time of falling. 



37,500 



31.57 H. P. Ans. 



33,000 X .035995 
(437) 127 -r- 62.5 = 2.032 = specific gravity. Ans. 



(438) 



62.5 

1,728 



X 9.823 = .35529 lb. Ans. 



(439) Use formula 21. ^=(^37). 

or 2 6 X 5 6 ^ * 7 6 5 5 X .48 = 499.2 lb. Ans. See l 
Fig. 37. 



(440) See Art. 961. 

'*G*»)-£-r 



5 X 25 3 



X 32.16' 



or 



1.5 X 25* 



/7 = 2x3 ; 16 = 466.42 lb. Ans. 
f -T- 12 

(441 ) (a) 2,000 — 4 = 500 = wt. of cu. 
ft. 500 -r- 62.5 = 8 = specific gravity. Ans. 



500 




.28935 1b. Ans. 




Fig. 37. 



(442) See Fig. 38. 14.5x2 
= 29. 30 X 29 = W X 5, or W = 

2^ = 174 lb. Ans. 



(443) 75 X .21 = 15.75 lb. 

Ans. 



Fig. 38. 

(444) ( a ) 900 X 150 = 135,000 ft. -lb. Ans. 
135,000 



(*) 



15 

9,000 



= 9,000 ft. -lb. per min. Ans. 



33,000 11 



-^H.P. Ans, 



ELEMENTARY MECHANICS. 



185 



(445) 900 X .18 X 2 = 324 lb. = force required to over- 
come the friction. 900 -f 324 = 1,224 lb. = total force. 

1,224 X 150 =>3709lH>Pt Ans> 
15 X 33,000 



(446) 18 ^ 88 = .2045. Ans. 



(447) See Art. 962. D = 



(448) See Fig. 39. 125 
-47.5 = 77.5 lb. = down- 
ward pressure. 

77.5-^4 = 19.375 lb. 
s= pressure on each support. 

Ans. 

(449) See Fig. 40. 



1,200 



W 



gV 32.16X3 



= 12.438. 



Fig. 39. 



Fig. 40. 

(450) See Fig. 41. 4.5 4-2 
12 



2.25 



X 6 X 30 



2.25. 
960 lb. Ans. 



(451) (a) 960-30 = 32. Ans. 

(b) 790 -f- 960 = .82^--= 82.29<i Ans 



Ans. 





186 



ELEMENTARY MECHANICS. 



;452) (a) See Fig. 42. 475 + (475 X .24) = 589 lb. 



Ans. 



(b) 475 ~ 589 = .8064 = 80.64$. Ans. 





Fig. 41. Fig. 42. 

(453) (a) By formula 23, U= FS= 6 X 25 = 150 

foct-poimds. Ans. 

w 4 sec - = t = fe min - 

Using formula 24, Power = ^ = ^|5 = 3,600 ft. -lb. per 

■* ST 

min. Ans. 



HYDRAULICS 

(QUESTIONS 454-503.) 



(454) The area of the surface of the sphere is 20X20X 
3.1416 = 1,256.64 sq. in. (See rule, Art. 817.) 

The specific gravity of sea-water is 1.026. (See tables of 
Specific Gravity.) The pressure on the sphere per square 
inch is the weight of a column of water 1 sq. in. in cross- 
section and 2 miles long. The total pressure is, therefore, 

1,256.64 X 5,280 X 2 X .434 X 1.026 = 5,908,971 lb. Ans. 

(455) 125 — 83.5 = 41.5 lb. = loss of weight in water = 
weight of a volume of water equal to the volume of the 
sphere. (See Art. 987.) 1 cu. in. of water weighs .03617 
lb.; hence, 41.5 lb. of water must contain 41.5^.03617 = 
1,147.4 cu. in. — volume of the sphere. Ans. 

(456) According to Art. 988, the height to which 
water will rise in a capillary tube varies inversely as the 
diameter of the tube; hence, the required height is found 
from the proportion 

x : 1 = ¥ V : J, 
from which we have 

x = Y \ u = A". Ans. 

(457) From the table of Coefficients for Circular Ver- 
tical Orifices, we find the coefficient for an orifice .60 foot 
in diameter to be .596, and for a diameter of 1.00 foot under 
the same head the coefficient is .594; the coefficient, there- 
fore, decreases .596 — .594 = .002 for an increase of .4 foot 
in the diameter of the orifice. For an increase of .1 foot 
the decrease in the coefficient is .002 -r- 4= .0005, and for 

For notice of the copyright, see page immediately following the title page. 



J 88 HYDRAULICS. 

an increase of .15 foot the decrease is .0005 X 1.5 = .00075; 
consequently, the coefficient for an orifice .75 foot in diam- 
eter is .596 — .00075 = .59525. 

Applying formula 34tf, Art. 996, we find the discharge 
to be 

g=6.299x.75 2 X.59525x |/20=9.432 cu. ft. per sec. Ans. 

(458) When 1 cubic foot of water per second flows 
through the pipe, the velocity of flow is 

1 

v= ^-= „ ntf . -~ = 5.093 ft. per sec. 

A . ( So4 X .0" r 

From the table of Coefficients of "Friction f for Smooth 
Iron Pipes, we find for a 6-inch pipe, f — .0226 with a veloc- 
ity of 5 feet per second, and / = . 0220 with a velocity of 6 feet 
per second; consequently, the decrease in the coefficient 
for an increase in velocity of 1 foot per second is .0226 
— .0220 = .0006, and for an increase of .093 foot per second 
the decrease is .0006 X .093 = .0000558, and the coefficient 
for a velocity of 5.093 feet per second is .0226 — .0000558 = 
.0225442, say .0225. 

From formula 45^, Art. 1026, the head required to 
produce the flow is 

k = -0^X5000X0.093- ^ x . ^ = ^ ^ 
64. o2 X .o 

This value represents all that portion of the total head 
which is absorbed in overcoming the frictional resistances 
to flow through the pipe and in giving the water its velocity 
of flow v. The pressure equivalent to the head thus 
absorbed is 

/> = .434:/i= .431 x 91.34= 39.64 lb. per sq. in. 
Since the head that remains as pressure in the pipe when 
the water is flowing through it is the difference between the 
head while the water is at rest and the head absorbed 
in overcoming the frictional resistances and giving the 
water its velocity of flow v, the pressure during flow is 
150 — 39.64= 110.36 pounds per square inch. Ans. 



HYDRAULICS. 189 

(459) The head corresponding to the pressure of 110 
pounds per square inch is 110 -f- .434 = 253.45 feet. With 
a coefficient of velocity of .98 the water issues from the 
nozzle with a velocity 

v— .98 X 8.02 4/253.45 = 125.13 feet per second. 

(See formula 32, Art. 994.) 

Neglecting all resistances, this water would rise to a 
height 

h = -f- = ^^ = 243. 4 feet. Ans. 
%g 64.32 

(460) (ft) The head on the center of the orifice is 5 feet 
plus 4 inches = 5J- feet, and the depth of the orifice is 
8 inches = f- foot. 

From the table of Coefficients for Rectangular Vertical 
Orifices, the coefficient for an orifice .5 foot deep under a 
head of 4 feet is .614, and under a head of 6 feet the coeffi- 
cient is. 609; the decrease for 1 foot is : — - = .0025, 

and for 1% feet the decrease is .0025 X 1^= .0033. The 
coefficient for an orifice .5 foot deep under a head of 5^- feet 
is, therefore, .614- .0033 = .6107. 

In the same way we find the coefficient for an orifice . 75 foot 

deep under a head of 5^ feet to be . 609 — ( — — — 

.6057. 

Finally, to find the value of the coefficient for an orifice 
§ foot deep, we have the difference for a difference in depth 
of .25 foot = .6107 - .6057 = .005, the difference for a differ- 
ence in depth of .01 foot is .005 ~ 25 = .0002, and the 
difference for a difference in depth of f — \ = .67 — .5 = .17 
foot is .0002 X 17 = .0034. From this we find the coefficient 
for an orifice § foot deep under a head of 5J feet to be 
.6107- .0034 = .6073. 

Applying formula 34*/, we have the discharge 

2 = 8.02 X .6073 X 1 XfX 4/5J = 7.499 cubic feet per 
second. Ans. 



x n 



190 HYDRAULICS. 

(b) Since 1 cubic foot = 7.48 gallons, the discharge in gal- 
lons per hour is 

7.499 X 7.48 X 3,600 = 201,930 gal. Ans. 

(461) See Art. 999. Because the coefficients of dis- 
charge for an orifice near the side or bottom can not be 
accurately determined. 

(462) The velocity with which the water leaves the noz- 
zle is v = .98<\/%g/i, and according to formula 38^, Art. 

,-.2 
IOI 2, the kinetic energy of a iet is K = W- — . Substitu- 

ting the value of v, we have, since 1 cubic foot of water 
weighs 62.5 pounds, 

K= JVx .98 2 X// = 3X 62.5 X .98 2 X 75 = 13,505.6 foot- 
pounds per second. Ans. 

(463) See Art. 990. 



v = \/%gh — 4/2 x 32.16 X 10 = 25.36 ft. per sec. Ans. 

(464) (b) Area of top or bottom of cylinder equals 
20 2 X .7854 = 314.16 sq. in. Area of cross-section of pipe = 
(f) 2 X .7854 = .1104 sq. in. 25 lb. 10 oz. = 25.625 lb. 
25.625 -f- .1104 = 232.11 lb., pressure per square inch on top 
or bottom exerted by the weight and piston. 

Pressure due to a head of 10 ft. = .434 X 10 = 4.34 lb. per 
sq. in. 

Pressure due to a head of 13 ft. = .434 X 13 = 5.64 lb. per 
sq. in. 

(Since a column of water 1 ft. high exerts a pressure of 
.434 lb. per sq. in.) 

Pressure on the top = pressure due to weight -f- pressure 
due to head of 10 ft. =232.11 + 4.34=236.45 lb. per sq. 
in. Ans. 

{a) Pressure on bottom = pressure due to weight -j- pres- 
sure due to head of 13 ft. = 232.11 + 5.64 = 237.75 lb. per 
sq. in. Ans. 

(c) Total pressure, or equivalent weight on the bottom = 
237.75 X 314.16 = 74,691.54 lb. Ans. 



HYDRAULICS. 191 

(465) .434 X li '== .051 lb., pressure due to the head of 
water in the cylinder at the center of the orifice. 

236.45, pressure per square inch on top, -j- .651 = 237.101, 
total pressure per square inch. Area of orifice = l 2 X . 7854 = 
.7854 sq. in. 

.7854 X 237.101 = 186.22 lb. Ans. 

(466) (a) Use formulas 27^ and 27. Sp. Gr. = 
w 11.25 



(W-IV')-(W-IV^) (91.25 -41)-(16 X 5 - 3* X 16) 
.556. Ans. 

w s p- Gr - = tf5p = do x 5 ) - ( 8 ?xi6) = 2 - 667 Ans 

(467) (a) The length of the weir is 4 feet 3 inches = 
4.25 feet. From the table of Coefficients of Discharge for 
Weirs with End Contractions, the coefficient for a weir with 
a head of .30 foot is .619 when the length is 3 feet and .621 
when the length is 5 feet; therefore, the coefficient for a 

length of 4.25 feet is .619 + / - 621 ~ - 6 1 j x L25 \ __ >62 025. 

In the same way, the coefficient for a weir under a head of 

.40 foot and 4.25 feet long is .613 + / - 615 ~ - 6 3 j x 125 \ _ 

.61425. Finally, the coefficient for a Aveir 4.25 feet long 
under a head of .38 foot is .62025 — (.62025 — .61425) X .8 = 
.61545; or, using but three decimal places, .615. 

Applying formula 37#, Art. 1006, the discharge is 
Q'= 5.347 X .615 X 4.25 X .38 f = 3.274 cu. ft. per sec. Ans. 

(b) The discharge in gallons per 24 hours is 
3.274 X 24 X 3,600 X 7.48 = 2,115,700 gallons. Ans. 

(468) From formula 48, Art. 1032, the slope is 

5 = ^=.0025, 

and, from formula 49, the hydraulic radius is 

_a _ 6.2832 _ 
r ~J ~ (T2832 - ' 



192 HYDRAULICS. 

From the table of Values of the Coefficient of Rough- 
ness, the coefficient for smoothly plastered masonry is 
found to be .011. Substituting the above values in Kutter's 
formula, Art. 1033, the value of c becomes 

23 + -U - 00155 



■OH ' -0025 1A1 .. 

From formula 50, Art. 1033, the velocity of flow is 



v = 141.06 \/l X .0025 = 7.053 ft. per second. 
The discharge is 

Q = A v = ° X 7.053 = 44.31 cu. ft. per sec. Ans. 

(469) Assuming a coefficient of velocity of .02 and ap- 
plying formula 47tf, Art. 1028,the approximate diameter 
of the pipe is 

rf= 0.479 ( • 03x y xg ') *= 1.014 feet. 

The velocity of flow for a pipe 1 foot diameter when dis- 
charging 5 cubic feet per second is 

5 
v = w ^ , = 6.366 feet per second. 
.78o4 

From the table, the coefficient f for a pipe 1 foot in diame- 
ter with a velocitv of flow of 6.366 feet per second is .0200 

-( ■ 0200 ;- 0190 )x. 366 = .0198. 

Using this value of /"in formula 47^, we have 

d= om (.0198 X 7,640 X6y =10U feet) neady ^ 

Since this differs but .003 foot from the result obtained by 
the first trial, it may be regarded as correct, and, con- 
sequently, a 12-inch pipe may be used. Ans. 



HYDRAULICS. 



193 



(470) (a) See Fig. 43. Area of cylinder =19 J 
pressure, 90 pounds per sq, in. Hence, the total 
on the piston is 19 2 X .7854 X 90 = 
25,517.6 lb. = the load that can be 
lifted. Ans. 

(b) The diameter of the pipe has 
no effect on the load which can be 
lifted, except that a larger pipe will tor™™*-. 
lift the load faster, since more water 
will flow in during a given time. 

(471) The head on the orifice 
corresponding to a pressure of 28 
pounds per square inch is 28 -f-.434 = 
64.5 feet. 

From the table of Coefficients for 
Circular Vertical Orifices the coeffi- 
cients for orifices . 20 foot and . 60 foot 
in diameter are found to be the same 
for high heads. For a head of 50 feet 
the coefficient is . 594, and for 100 feet 



X .7854; 
pressure 




Fig. 43. 



of 64.5 feet the coefficient is .594 



592 ; hence, for a head 
594 - .592 N 



/ .594 - .592 X 

V"~6o-"7 



X 14.5 



.5934. 

Applying formula 34#, Art. 996, the discharge is found 
to be 

Q = 6.299 X .25 2 X .5934 X 4/645 = 1.876 cu. ft. per second. 

Ans. 

(472) (a) 36 in. = 3 ft. A column of water 1 in. square 
and 1 ft. high weighs .434 1b. .434x43 = 18.662 lb. per 
sq. in., pressure on the bottom of the cylinder. .434 X 40 = 
17.36 lb. per sq. in., pressure on the top of the cylinder. 
Area of base of cylinder = 20 2 X .7854 = 314.16 sq. in. 
314.16 X 18.662 = 5,862.85 lb., total pressure on the bottom. 

Ans. 






(6) 314.16 X 17.36 
top. Ans. 



5,453.82 lb., total pressure 



the 



124 



HYDRAULICS. 



(473) 2 lb. = 32 oz. 32 — 10 = 22 oz. = loss of weight 
of the bottle in water. 32 -+- 1G = 48 = weight of bottle and 
sugar in air. 48 — 16 = 32 oz. = loss of weight of bottle 
and sugar in water. 32 — 22 = 10 oz. = loss of weight of 
sugar in water = weight of a volume of water equal to the 
volume of the sugar. Then, by formula 27, 

W 16 

10 



specific gravity = 



(474) ZZ = V%gk 
33 2 



h 



64.32 



W-W 
(see Art. 990), 
16.931 ft. Ans. 



1. 



Ans. 



or 



(475) The head h that produces the flow through the 
opening is the difference in level of the water on the two 
sides of the partition, or h — 8 — 3£ = 4J- feet. 

Applying formula 34^, Art. 999, we have 

Q= .Ql5bd\/¥gh = .615 X .75 2 X 8.02 |/£]f= 5.885 cu. ft. 
per second. Ans. 

(476) First find the mean velocity of flow for each 
section. To do this divide the distance between the wires 
in feet by the time in seconds for each section; then multi- 
ply the velocity for each section by the area of that section, 
thus finding its discharge. The sum of the quantities dis- 
charged by the different sections will be the discharge of 
the stream. The results should be tabulated about as 
follows: 

Section Time. Velocity in Discharge in 



No. 


m. 


sec. sec. 


Feet 


per Second. 


Cubic Feet per Second. 


1 


. 3 


24 =204 . 


.300- 


-204 =1.470. 


.1.470x2.36= 3.469 


2 


. 2 


32 =152 . 


.300- 


-152 =1.974 . 


.1.974x5.74= 11.331 


3 


. 1 


15. = 75 . 


.300- 


- 75 =4.000. 


.4.000x7.48= 29.920 


4 


. 


m= 87*. 


.300- 


- 37^=8.000. 


.8.000x9.78= 78.240 


5 


. 


40 = 40 . 


.300- 


- 40 =7.500. 


.7.500x9.57= 71.775 


6 


. 


57 = 57 . 


.300- 


- 57 =5.263. 


.5.263x8.24= 43.367 


7 


. 1 


8^= 68*. 


.300- 


- 68^=4.379. 


.4.379x7.15= 31.310 


8 


. 2 


45 =165 . 


.300- 


-165 =1.818. 


.1.818X4.09= 7.436 


9 


. 3 


3 =183 . 


.300- 


-183 =1.639 . 


.1.639x3.35= 5.491 


10 


. 3 


47 =227 . 


.300- 


-227 =1.321 . 


.1.321x1.98= 2.616 




Total 284.955 cu. ft 












per second. Ans. 



HYDRAULICS. 



195 



(477) {a) 1 cu. in. of water weighs .03017 lb. 
.03017 X 40 = 1.4408 lb. = weight of 40 cu. in. of water = 
loss of weight of lead in water. 

10.4 — 1.447 = 14.953 lb. weight of lead in water. Ans. 

(b) 16.4 -f- 40 = .41 = weight of 1 cu. in. of the lead. 
10.4 — 2 = 14.4 lb. = weight of lead after cutting off 2 lb. 
14.4 -s- .41 = 35.122 cu. in. = volume of lead after cutting 
off 2 lb. Ans. 



(478) (a) See Fig. 44. 13.5" X 9 X .7854= 95.4201 sq. 
in., area of base. 

.03017 X 20 = .7234 lb. per sq. in., pressure on the base 
due to the water only. 

12 +.7234= 12.7234 lb., 
total pressure per square 
inch on base. 

12,7234 X 95.4201 = 
1,214.144 1b. Ans. 

(b) 47 X 12 = 504 lb., 
total upward pressure. 
Ans. 



(479) (a) See Fig. 44. 
4 sin 53° = 3.195", nearly. 

20 - 3.195 = 16. 805" = 
distance of center of gravity 
of plate below the surface. 

.03017 X 10.805 + 12 = 
12.00784 lb. per sq. in. = 




Fig. 44. 

perpendicular pressure against the plate. 5 X 8 = 40 sq. 
area of plate. 

12.00784 X 40 = 504.314 lb. = perpendicular pressure 
plate. Ans. 

(b) 504.314 sin 53° = 402.70 lb. = horizontal pressure 
plate. Ans. 

(c) 504.314 cos 53° = 303.5 lb. = vertical pressure 
plate. Ans. 

G. G. IV.— 12 



m. 



on 



on 



196 HYDRAULICS. 

5 2 X 7854 
(480) =-j-j = area of pipe in square feet. Using 

formula 28^, 

Q = A v m = '- — '— x 7.2 = discharge in cu. ft. per sec. 

X 7.2 X 7.48 = discharge in gal. per sec. 



144 

5 2 X .7854 



7.2 X 7.48 X 60 X 60 X 24= 634,478 gal. dis- 



144 

charged in one day. Ans. 

(481) Area of &|~in. circle = 4.9087 sq. in.; area of a 
2-in. circle = 3.1416 sq. in. (4.9087 - 3.1416) X 12 = 21.2052 
cu. in. of brass. 

21.2052 X .03617 = .767 lb. = weight of an equal volume 
of water. 

6 lb. 5 oz. =6.3125 lb. 6.3125 -f- .767 = 8.23 Sp. Gr. of 
brass. Ans. 

(482) (a) The wetted perimeter, that is, the length of 
the boundary of a section exposed to the water, is 

8 + 2 X 5 -s- sin 30° = 8 + 2 X -z = 28 feet. Ans. 

. o 

(J?) Since the length of each of the slanting sides of a 

section is 10 feet, and the depth of the water 5 feet, the 

length of the top of a section through the water is 

8 + 24/IO 2 — 5 2 = 25.32 feet. The area of the section is 

95 32 -1-8 * 

— '' X 5 = 83.3 square feet. From formula 49, Art. 

2 

1032, the hydraulic radius is 

r = - = — — = 2- 975 feet. Ans. 
/ 2b 

(483) (a) From the table giving the values of the 
coefficients of roughness n for various channels (see Art. 
1033), we find the coefficient for well-made earth canals to 
be .0225. The slope of the canal is 

S = 7= i,) 2 ' 2 o <nr> = • 0000 " 8 ( see formula 48, Art. 1 032). 



HYDRAULICS. 197 

From the last example, we have the hydraulic radius r = 
2.975 feet. Substituting these values in Kutter's formula, 
we have 

03 , 1 I -00155 

^ .0225 " r . 000078 
c = ; -tttz^^i — = 78. 55. 



.6681 + ^88 + .000078; ^975 
From formula 50, Art. 1033, we have 



v=c\/r s= 78. 55 4/2. 975 X. 000078 = 1.196 feet per sec. Ans. 

(b) According to the solution of the second part of the 
last example, the area of the section is 83.3 square feet; 
therefore, the discharge is 

Q = av= 83.3 X 1.196 =.99.627 cu. ft. per sec. Ans. 

(484) The discharge in cubic feet per second is 

5,000,000 _ 773:cu ft 

g -7.48x24x60x60- 7 ' 7d7CU - "' 

Since the area of a 24-inch pipe is 3.1416 square feet, the 
velocity of flow will be 

7 737 
v = ^-tJTa = 2-463 feet per second. 

The head required to produce a velocity of flow of 2.463 
feet per second through a 24-inch pipe is found by applying 
formula 45#, Art. 1026. First find the value of /for the 
given diameter and velocity from the table of Coefficients 
for Smooth Iron Pipes as follows: 

/for a 24-inch pipe with a velocity of 2' per second = .020 
/ for a 24-inch pipe with a velocity of 3' per second = .019 

Difference for 1' velocity = .001 
Difference for .463' velocity = .000463 

.020— .000463 = .019537, or, using the nearest decimal to 

four places, .0195. 

Substituting the given values in the formula, 

, .0195 X 5,362 X 2.463" , no9Qv0 , 10J K A1V , 
h = 64 g 2 2 • + .0233 X 2.463 2 = o.07 feet. 



198 HYDRAULICS. 

The total head against which the pump must work is 
375 + 5.07 = 380.07 feet. 

380.07 X .434: = 1G4. 95, say 105, pounds per square inch, 

Ans. 

(485) The area of a 48-inch pipe is 12.5664 square feet; 
therefore, the velocity of flow is 

80 
v — ^ r , ftr t = 6.366 ft. per sec. 
12.5664 

From the table of Coefficients f for Smooth Iron Pipes, 
the value of /for a 48-inch pipe with a velocity of 6.366 ft. 
per sec. is found as follows: 

/ for a 48-inch pipe when v = 6 f t. per sec. = .012 

/ for a 48-inch pipe when v = 8 ft. per sec. = .0115 

Difference for an increase of 2 ft. per sec. = .0005 
Difference for an increase of 1 ft. per sec. = .00025 
Difference for an increase of .865 ft. per sec. = .00025 X .366 = 

.00009. 

/ for 48-inch pipe when v = 6.366 ft. per sec. = .012 - .00009 = .01191, 
say .0119. 

The head required to produce the flow is found by apply- 
ing formula 45, Art. 1026. The value of m to be used 
is .49 (see Art. 1020), and, according to Table 18£, Art. 
1 023, c' = .294 for a bend whose radius is equal to twice 
the radius of the pipe. Substituting these values in the 
formula, 

. 6.366 s (1 + .0119 X *£*-+. 49 + .294) . „ , . . 

h = — — — ^ — ! = 1.62 feet. Ans. 

64.32 

(486) Use formula 46£, Art. 1027; whence, sub- 
stituting values, 

Q = .0408 X 6 2 X 7.5 = 11.016 gal. per sec. Ans. 

(487) 14 2 X .7854 X 27 = volume of cylinder = volume 
of water displaced. 

14 2 X .7854 X 27 X .03617 = 150 lb., nearly, = weight of 
water displaced. 

(14 2 — 13 2 ) X .7854 X 27 — volume of the cylinder walls. 



HYDRAULICS. 199 

13 2 X .7854 X £ X 2 = volume of the cylinder ends. 
.261 lb. = weight of a cubic inch of cast iron; then, 
[(14 2 - 13 2 ) X .7851 X 27 + 13 2 X .7854 X ± X 2] X .261 = 
167 lb., nearly, == weight of cylinder. Since weight of 
cylinder is greater than the weight of the water displaced, 
it will sink. Ans. 

(488) 2 lb. - 1 lb. 5 oz. = 11 oz., weight of water. 
1 lb. 15.34 oz.— 1 lb. 5 oz. = 10.34 oz., weight of oil. 
10.34 -r-.il = .94 = Sp. Gr. of oil. Ans. 

(489) The length of the weir in feet and decimals is 
5.292, or 5.3 feet, nearly. From the table of Coefficients 
for Weirs Without End Contractions, we find the coefficient 
for a weir 5.292 feet long under a head of .43 foot as follows: 

Coefficient for head of .40 foot and length of 5 feet = .628 
Coefficient for head of .40 foot and length of 7 feet = .625 



Difference for increase of 2 feet in length = .003 
Difference for increase of 1 foot in length = .0015 
Difference for increase of .3 foot in length = .00045 

Coefficient for head of .40 foot and length of 5.3 feet = 
.628- .00045 = .62755. 

In the same way, the coefficient for a weir 5.3 feet long 
under a head of .50 foot is found to be .62655. The differ- 
ence for an increase in the head of .10 foot is .62755 
— .62655 = .001, the difference for an increase of .03 foot is 
.001 X .3 = .0003, and the coefficient for a weir 5.3 feet long 
under a head of .43 foot is .62755 — .0003 = .62725, or, using 
but three decimal places, .627. 

Applying formula 36^, Art. 1006, we have the dis- 
charge 
Q' = 5. 347 X . 627 X 5. 292 X . 43 f = 5. 00 cu. ft. per sec. Ans. 

(490) According to Art. 1008, the coefficient of dis- 
charge for a standard tube may be taken as .82, and the 
coefficient of velocity is the same as the coefficient of dis- 
charge. Hence, the velocity is 

v = .8# X 8.02 X |/64 = 52.61 ft. per sec. Ans. 



200 HYDRAULICS. 

(491) (b) The velocity of flow in the pipes is inversely 
proportional to their areas, and, since the areas are propor- 
tional to the squares of the diameters, the velocity of flow 
is inversely proportional to the squares of the diameters of 
the pipes. Letting v — the velocity in the 6-inch pipe, 
we have 

v : 2 = 18 2 : 6 2 , 
from which v = 18 feet per second. Ans. 

(a) The total head in the 18-inch pipe is the pressure 

head, 50 pounds per square inch, and the velocity head 

v 2 4 

which is h — — - = — — - = . 06219 foot. 
%g 64.32 

A head of .06219 foot is equivalent to a pressure of 

.06219 X .434= .027 pound per square inch, nearly. The 

total pressure head in the 18-inch pipe is, therefore, 50 -)- 

.027 — 50.027 pounds per square inch. 

18 2 
The velocity head in the 6-inch pipe is h = = 5.037 

64. o2 

feet, equivalent to 5.037 X .434= 2.186 pounds per square 
inch. 

Since no head is absorbed in overcoming frictional resist- 
ances, the pressure in the 6-inch pipe must be equal to the 
total pressure head in the 18-inch pipe minus the head 
equivalent to the velocity of flow in the 6-inch pipe; that is, 
to 50.027 — 2.186 = 47.841 pounds per square inch. Ans. 

(492) (a) According to Art. 1029, the effective head 
is the difference in level of the surface of the water in the 
two reservoirs. This gives us 

h = 42+27- 19 = 50 feet. 

Applying formula 44, Art. 1025, and using the approx- 
imate coefficient f ■= .02, we have the approximate velocity 
of flow 

v = 8.02 y r l X f** = 5.246 feet per second. 
r .02 X 5842 r 

From the table of Coefficients/" for Smooth Iron Pipes, the 
value of /"for a 12-inch pipe and a velocity of flow of 5.246 



HYDRAULICS. 201 

feet per second is .0200, very nearly. Therefore, the 
velocity just found may be taken as correct. Ans. 
(b) The discharge in gallons per 24 hours is 
Q = 5.246 X .7854 X 7.48 X 24 X 60 X 60 = 2,662,750 gal- 
lons. Ans. 

(493) The breadth of the orifice in feet is b = 7£ -4- 12 = 
-J!- foot, the head on the upper edge is h x = 2 feet, and the 
head on the lower edge is // 3 = 2 feet + ^j- feet = 3-J- feet. 

Substituting these values in formula 34^", Art. 998, the 
discharge is found to be 

g= 8.02 X .615 XiXUiVW) 3 - V% z ) = 5.771 cu. ft. 
per second. 5.77 X 60 = 346.25 cu. ft. per minute, nearly. 

Ans. 

(494) In this case the breadth of the orifice is b = \% = 
1\ feet, the head on the upper edge is 1i x = 2 feet, and the 
head on the lower edge is h 2 = 2 feet + -J-f foot = 2^f feet. 
Substituting in formula 34^, as in the last example, the 
discharge is 

Q = 8.02 X .615 X fX 1| (/(If) 3 - /2 3 ) =5.466 cu. ft. 
per sec. 5.466 X 60 = 328 cu. ft. per minute. Ans. 

(495) 6 X 4 X .7854 = 18.85 sq. in. = area of upper 
surface. 

15 2 X .7854 = 176.715 sq. in. = area of base. 

132 

X 176.715 = 1,237.5 lb., pressure due to weight on 

lo. oO 

upper surface. 

.03617 X 24 X 176.715 = 153.4 lb., pressure due to water 
in vessel. 

1,237.5 + 153.4 = 1,390.9 lb., total pressure. Ans. 

(496) A sketch of the arrangement is shown in Fig. 45. 
(a) Area of pump piston = (^-) 2 X .7854 = .19635 sq. in. 
Area of plunger = 10 2 X .7854 = 78.54 sq. in. 

Pressure per square inch exerted by piston = — - lb. 

.19655 

Hence, according to Pascal's law, the pressure on the 
plunger is -j^ X 78.54 = 40,000 lb. Ans. 

. J. JuOO 



202 



HYDRAULICS. 



(/;) Velocity ratio = 11 : .00375 =400 : 1. Ans. 
(c) According to the principle given in Art. 977, P X 
\\ inches = If X distance moved by plunger, or 100 X 



100 lb. 

1 




Fig. 45. 

1-J = 40,000 X required distance; hence, the required dis- 

100 y 11 

tance = * p 2 = .00375 in. Ans. 

(497) (a) See Art. 986. 

W =2 lb. Si'oz. =40.5 oz. 
w = 12 oz. 

i>F' = 1 lb. 11 oz. = 27 oz. 

By formula 27c, 

IV -w 40.5-12 28.5 



Sp. Gr. 



= 1.9. Ans. 



W — w ' 27-12 ' ' 15 
(b) 15 oz. = f|lb. = .93751b. .9375 — .03617 = 25.92 cu. 



HYDRAULICS. 203 

in. = volume of water = volume of slate. Therefore, the 
volume of the slate = 25.92 cu. in. Ans. 

(498) (a) 4 ft. 9 in. = 4.75 ft. 19 - 4.75 = 14.25. 
Range = \/kky = 4/4 X 4.75 X 14725 = 16.454 ft. Ans. 
(I?) 19 — 4.75 = 14.25 ft. Ans. 

(c) 19 -h 2 = 9.5. Greatest range = 4/4 X 9J5 2 = 19 ft. 
Ans. (See Art. 992.) 

(499) Assuming a value of the coefficient f= .02, and 
substituting in formula 44#, Art. 1025, we have the ap- 
proximate velocity of flow 



v = 2.315 V If * ™ = 11.56 ft. per sec. 
r .02X6,570 r 

From the table of Coefficients f for Smooth Iron Pipes,* 
the value of /"for a 10-inch pipe and a velocity of 11.5 feet 
per second is found to be nearly .0185. Substituting in the 
formula again, using this value of/", we have 

^ = 2 - 315 ^iSS!?ro =13 - 04ft -P ersec - . 

The value of f corresponding to this velocity is found to 
be .0182, which, substituted in the formula, gives 



-^ 15 ^ 0l5^70 = 12 - laft -P erSeC - 

This corresponds to a value of f of .0183, which, substi- 
tuted, gives 

^= 3 - 3i5 ^SI|S= i2 - o9ft -p ersec - ■ 

This corresponds so closely with the last value of v, it may 
be assumed to be correct. 

Applying formula 46$, the discharge in gallons per 
second is 

Q = .0408 X 10' 2 X 12.09 = 49.33 gal. per sec. Ans. 

(500) The pressure head at any point in the pipe is 
equal to the hydrostatic head at that point minus the 



204 HYDRAULICS. 

various losses of head in the pipe between the reservoir and 
the given point. (See Arts. 1018 to 1027.) 

The head absorbed in producing a velocity of flow of 
12 feet per second (see example 499) is 

v 2 144 

A ' = ^.= FT^>- = 2 - 239feet - 
%g 64.32 

The head absorbed in overcoming the friction in the pipe 
is, according to formula 40z, Art. 1021, 

/r =f± f- = .0183 X ^ X ^ = 98.33 feet. 
d %g | 64.32 

The head absorbed in overcoming the resistances at the 
entrance to the pipe is found by applying formula 39, Art. 
1020, 

h'" = ^- = .5X Ss = 1.H9 feet. 

The total head absorbed is, therefore, 

h'-\- #"'+ k iv = 2. 239 + 1. 119 + 98. 33 = 101. 688 feet. 

The hydrostatic head is 150' feet; hence the pressure head 
is 150 — 101.688 = 48.312 feet, and the pressure is 

48.312 X .434 = 21 pounds per square inch, nearly. Ans. 

(501) («) The velocity of flow through a 1-inch pipe 
when discharging 0.3 gallons per second is 

V = i = 7.48 X (A)' X .7854 = 7 - 353 f6et Per SeC ° nd " 

From the table of Coefficients f for Smooth Iron Pipes, 
we find the value of f for a 1-inch pipe with a velocity of 
7.353 feet per second to be .0260, very nearly. Ans. 

(b) The velocity is 

V = 7.48x(jSrx.7854 = 3 - 83 ° feet Per SeCOnd 

From the table, the value of f for this velocity and a 
4-inch pipe is .0243, nearly. Ans. 



HYDRAULICS. 205 

(c) The velocity is 

375 

v = 7854 6() = 7.958 feet per second. 

From the table we have 

/"= .0190 for this diameter and velocity. Ans. 

(d) The velocity is 

8,000,000 _ ..- ' , 

9 = 7.48 X 2- X. 7854 X 24x60x60 = ^ ^ ^ ^^ 

From the table we have 

f= .0171 for this diameter and velocity. Ans. 

(e) The length of the pipe in feet is 3£ X 5,280 = 17,160 
feet. 

Applying formula 44, Art. 1025, using the approxi- 
mate value of the coefficient f of .015, the approximate 
velocity of flow is found to be 



J 



v = 8.02 y — - = 7.069 feet per second. 

• UJ-0 X J-/, -Lou 

The value of f corresponding to this velocity is .0117, 
which, substituted in the formula, gives 



v = 8.02 / 011 y*n,l60 = 8 - 004 feet P er second - 

This gives us a value of /"—.0115. Substituting this 
and solving, we have 



v- 



V=8 -° 2V .0115X17,160 = 8 '° n f6et Per S6COnd ' 

a value that corresponds almost exactly with the value found 
before; hence, we have f = .0115 as the correct value to be 
used. Ans. 

(502) Specific gravity of sea-water is 1.026. Total area 
of cube = 10. 5 2 X 6 = 661. 5 sq. in. 1 mile = 5,280 ft. Hence, 
total pressure on the cube = 661.5 X 5,280 X 3.5 X .434 X 
1.026 = 5,443,383 lb. Ans. 

(503) 19 2 X .7854 X 80 = 22,682 lb. Ans. 



PNEUMATICS. 

(QUESTIONS 504-553.) 



(504) The force with which a confined gas presses 
against the walls of the vessel which contains it. 

(505) (a) 4 X 12 X .49 = 23.52 lb. per sq. in. Ans. 
(b) 23.52 -T- 14.7 = 1.6 atmospheres. Ans. 

(506) (a) A column of water 1 foot high exerts a pres- 
sure of .434 lb. per sq. in. Hence, .434 X 19 — 8.246 lb. per 
sq. in. , the required tension. A column of mercury 1 in. high 
exerts a pressure of .49 lb. per sq. in. Hence, 8.246 -r- .49 = 
16.828 in. = height of the mercury column. Ans. 

(&) Pressure above the mercury = 14.7 — 8.246 = 6.454 
lb. per sq. in. Ans. 

/**VT\ TT • r i *o j, P v (14.7 X 3) X 1 

(507) Using formula 53, p^= r — = ± —- 1 = 

17.64 lb. Ans. 

(508) (c) Using formula 61, 

„ .37052 WT .37052X7.14X535 .. 1QQ -. . 

V= • = ■ kzt^ = 64.188 cu. ft. Ans. 

p 22. 05 

(T= 460° + 75° =535°, and/ = 14.7 X 1.5 = 22.05 lb. 
per sq. in.) 

(a) 7.14 -7- .08 = 89.25 cu. ft., the original volume. Ans 

(b) If 1 cu. ft. weighs .08 lb., 1 lb. contains 1 -i- .08 = 
12.5 cu. ft. Hence, using formula 60, p V— .37052 T, or 

T = ~£Lt = 22 '°i^o 2 ' 5 = 743.887°. 743.887 - 460 = 
283. 887 \ Ans. 

For notice of the copyright, see page immediately following the title page. 



208 PNEUMATICS. 

(509) Substituting in formula 59, / = 40, t - 120, and 

t = 55 

„ \n/ 460 + 55 \ 40X515 „-«„« A 

^> = 40 (ieo + iSo) = ~^8o- = 3o - 517 lb - Ans ' 

(510) Using formula 61,/ ^= .37052 0T, or 

^=ST- r=460 ° + 60 ° = 520 °- 
Therefore, ^= 3^^^ =.076896 lb. Ans. 

(511) 175,000 -^- 144 = pounds per sq. in. 

(175,000 -r- 144) -r- 14.7 = 82.672 = atmospheres. 

Ans. 

(512) Extending formula 63 to include 3 gases, we have 

/ > P = A*',+A*'.+A*'.. or 40xP-= 1 X 12+ 2X10 + 3X8. 

Hence, P=— = IA atmos. = 1.4 X 14.7 = 20.58 lb. per 
40 

sq. in. Ans. 

(513) In the last example, PV=56. In the present 
case, P= — — atmos. Therefore, V — -p = —— = 35.79 
cu. ft. Ans. JA7 

(514) For / = 280°, T=740°; for t = 77°, T=53T. 
p V=. 37052 WT y or W= ^^ j- (Formula 61.) 

w • u* c u , • 14.7 X 10,000 co „ ,_ lu 
Weight of hot air = ^ x UQ = 536.13 lb. 

w • 1,4. * • a- i ^ 14.7X10,000 M001t , 
Weight of air displaced = — '—- = 738.81 lb. 

.o7UO/d X 5o7 

738. 81 - 536. 13 = 202. 68. 202. 68 - 100 = 102. 68 lb. Ans, 

(515) According to formula 64, 





PNEUMATICS. 209 

Of) \y Q1 

Therefore, T= ^ * To =439.35°. Since this is less 
1.4111bo 

than 460°, the temperature is 460 — 439.35 = 20.65° below 

zero, or — 20.65°. Ans. 

(516) A hollow space from which all air or other gas 
(or gaseous vapor) has been removed. An example would 
be the space above the mercury in a barometer. 

(517) One inch of mercury corresponds to a pressure of 

.49 lb. per sq. in. 

1 .49 

— inch of mercury corresponds to a pressure of '-—z lb. per 

49 
sq. in. '— X 144 = 1.764 lb. per sq. ft. Ans. 

(518) (a) 325 X .14= 45.5 lb. = force necessary <ci 
overcome the friction. 6 X 12 = 72" = length of cylinder. 
72 — 40 = 32 = distance which the piston must move. Since 
the area of the cylinder remains the same, any variation in 
the volume will be proportional to the variation in the 
length between the head and piston. By formula 53, 

pv = pv y . Therefore, p =+-±—± = — = 8.1^- lb. per 

r j-i i ,r y 72 3 r 

sq. in. = pressure when piston is at the end of the cylinder. 

Since there is the atmospheric pressure of 14.7 lb. on one 

2 
side of the piston and only 8.1 — lb. on the other side, the 

o 

2 
force required to pull it out of the cylinder is 14.7 — 8.1- = 

o 

6.5- lb. per sq. in. Area of piston = 40 a X .7854 = 1,256.64 
o 

sq. in. Total force = 1,256.64 X 6.5^- = 8,210.05. Adding 
the friction, 8,210.05 + 45.5 = 8,255.55 lb. Ans. 

(b) Proceeding likewise in the second case, pv=p x v lt or 

■h v 1 4- 7 V 40 
p = ~f = 6 = 98 lb. 98 - 14.7 = 83.3 lb. per sq. in. 

1,256.64 X 83.3 -f 45.5 = 104,723.612 lb. Ans. 



210 PNEUMATICS. 

(519) S.47 = original volume = v v 8.47-4.5 = 3.97 
cu. ft. = new volume = v. By formula 53, 

= pv = 3.97 y : 38 = lb r . n> Ans> 

/l z/j 8.47 

(520) Original weight = IV = .5 lb. = 8oz. ; new weight 
= W t = 1 lb. 6 oz. = 22 oz. According to formula 56, 

/ ^ i =/i ^ ; or A = tt/ = » = 40.42o lb. per 

sq. in. Ans. 

(521) Applying formula 58, 

/4G0 + M 4,516 / 460 + 80 \ , ._ , . 

"-^4^)-1^8(460T^o) = 1 ' 96CU - ft - AnS ' 

(522) According to formula 61,/ V — . 37052 J>J/ 7, or 
' Fp = ^^ = U-7 X 1.25 X 55 = Ans 

.37C52 r .37052 X o48 

(523) Using formula 63, PV=pv +p x v xi or P x 7. 5 

= 14.7 X 2 X 7.5 + 40 X 7.5, or P= 69.4 lb. per sq. in. Ans. 

(524) 48% 36", and 24" = 4', 3', and 2', respectively. 
Hence, 4 X 3 X 2 = 24 cu. ft. = the volume of the block. 
The block will weigh as much more in a vacuum as the 
weight of the air it displaces. In example 510, it was found 
that 1 cu. ft. of air at a temperature of 60° weighed 
.076296 lb. .076296 X 24 + 1,200 = 1,201.83 lb. Ans. 

(525) (a) (See Art. 1088.) 127 + 16 = 143. 
(A) a X .7854X1X125X62.5X143 _ 

33,000 
14.9563 ~- .75 = 19.942 H.P. Ans. 
(b) Discharge in gallons per hour = volume of cylinder 
in cu. ft. X number of strokes per minute X 7.48 X 60 = 



(5)" 



X .7854 X 125 X 7.48 X 60 = 24,784.3 gal. per hr. Ans. 



(526) In this example, the number of times that the 

pump delivers water in 1 minute is 100 -f- 2 = 50; in the last 

example, 125. Hence, the number of gallons discharged 

50 
per hour in this case will be 24,786 X r^rr = 9,914.4 gal. Ans. 



PNEUMATICS. 211 

(527) See Art. 1043. 

gO 23 

Pressure in condenser = ■ — — ■ X 14.7 = 3.43 lb. per sq, 

60 . « 

in. Ads. 

(528) 144 X 14.7 = 2,116.8 lb. per sq. ft. Ans. 

(529) .27 + 3 = .09 = weight of 1 cu. ft. Using formula 
56, 

/ W x =p x W, or 30 W x = 65 X .09. W x = .195 lb. Ans. 

(530) Using formula 61 , 

/ V— .37052 W T, or 30 X 1 = .37052 X .09 X T. 

Q(~) 

T = ^JV „ A = 899. 6°. 899. 6° - 460° = 439. 6°. Ans. 
. 37052 X. 09 

(531 ) 460°+ 32° = 492° ; 460°+ 212° = 672° ; 460°+62° 
= 522°, and 460° + ( - 40°) = 420°. 

(532) Using formula 61,/ V=. 37052 W T, and 

substituting, 

14- 7y 10v4- 
(14.7X10) X 4 =. 37052X3.5 X T, or T= ~^^j^ = 

453.417°. 453.417 - 460° = - 6.583°. Ans. 

(533) Using formula 63, V P=vp-\-v x p„ we find 

D 15X63+19X14.7X3 



25 



71.316 lb. Ans. 



(534) Using formula 60, / V = .37052 T, or P = 

37052X540 on „ . 1 . 

- = 20 lb. per sq. in., nearly. Ans. 



10 

(535) One inch of mercury represents a pressure of 
.49 lb. Therefore, the height of the mercury column is 
12.5-v-.49 = 25.51 in. Ans. 

(536) Thirty inches of mercury corresponds to 34 ft. 
of water. (See Art. 1043.) Therefore, 

30' 7 : 34 ft. :: 27" : x ft., or x = 30.6 ft. Ans. 
A more accurate way is (27 X .49) -r- .434= 30.5 ft. 
a. a. i v.— 13 



212 PNEUMATICS. 

(537) {a) 30 — 17.5 = 12.5 in. = original tension of gas 
in inches of mercury. 30 — 5 = 25 in. = new tension in 
inches of mercury. 

VP=vp+v x p x (formula 63), or 6.7x25 = 6.7x12.5+^x30. 

6.7X25-6.7X12.5 . „_ 1 - . 

v. = ■ — = 2.79 - cu. ft, Ans. 

(b) To produce a vacuum of inches, 

6.7X30-6.7X12.5 . __ Q u . 
v x = — = 3. 908 cu. ft. Ans. 

(538) 11 + 25 = 36, final volume of gas. 2.4 -5- 36 = 

^ lb. Ans. 
15 

(539) Using formula 59, 

V460 + M _ /460 + 300X , w e . lu 

A = A-ieoT/) = 12 x (leo+w) = 17 ' 54 lb - per sq '| n ns 

(540) T= 460 + 212 = 672°. Using formula 61,/ P= 
.37052 ^r, we have 14.7 X 1 = .37052 X IV X 672, or W = 

14.7 



.37052 X 672 



,059039 lb. Ans. 



/ K/l1 , ,. 20 2 X. 7854x32 K ■__ . . 

(541) (a) -^— = 5.8178 cu. ft. = volume 

of cylinder. 

32 — 26 = 6 in., length of stroke unfinished. 

5.8178 X 577 = 1.0908 cu. ft. Ans. 

(b) By formula 61, taking the values of /, F, and T at 

the beginning of stroke, 

*ir o-AM rrr-r tj7 /^ 14.7X5.8178 

/F=. 37052 ^T, or ^_ ;^ T = "^052^35 = 

.43143 lb. Ans. 

(c) Now, substituting in formula 61 the values of V, 
W f and T at time of discharge, 

.37052 WT .37052 X .43143 X 585 OK wow „ 

/= v = L0908 = 85 - 72? 1K Per 

sq. in, Ans. 



PNEUMATICS. 213 

(542) Using formula 63, VP= vp+v t p iy or 30 X 35 
= 19 X 12 + 21 A, or /, = 30X35 ~ 19Xl2 = 39.14 lb. per 
sq. in. Ans. 

(543) Use formula 64. PV= U^ +&p>\r. 

T- 460 + 72 = 532. 
X 45 , 17 X 60> 



/ 13X 
\ 52 



532 



Therefore, P= v 52 ° — 54 ° ' = 26.723 lb. per 

sq. in. Ans. 

(544) See Art. 1043. Since the barometer column 
stands at 30 inches for a perfect vacuum, the height when 
the air is admitted will be 30" — 4£* = 25^". Ans. 

(545) Sp. Gr. of alcohol =.8. Therefore, 16 X. 434 X- 8 = 

pressure exerted by the column of alcohol. '—— — = 

11.337 in. = height of a column of mercury that will give 
the same pressure as 16 ft. of alcohol = number of inches 
shown by the gauge. Ans. 

(546) (a) 14.7 + 9 = 23.7 lb. per sq. in. Using 
formula 53, 

pv 14.7X80 ^ . 

^ = X = _ ^- = 49 ' 62m - = 

distance between piston and end of stroke. Since the area 
of the piston remains constant, the volume at any point of 
the stroke is proportional to the distance passed over by the 
piston. Hence, we may use the latter for the former in the 
formula. 80 — 49. 62 = 30. 38 in. Ans. 

(b) Area of piston = 80* X .7854. The volume of air at 
point of discharge is 80 2 X .7854 X 49.62 cu. in. = 

80 9 X .7854 X 49.62 



1,728 



= 144.34 cu. ft. Ans. 



(547) Using formula 56,/ W^p x W, or 3.5x147x2= 

A X 13; hence, p x = 14 - 7 X 3j_X _2 = 79l5+ lb per $q in 

Aius 



214 PNEUMATICS. 

(548) GO" — 50" = 10". Since the volumes are propor 
tional to the lengths of the spaces between the piston and 
the end of the stroke, we may apply formula 62, 

pV p,V y 14.7 X GO /, X 10 

- — — —^- — -• or — 

T T x ' 460 + GO 460 + 130* 

rnu c * 14.7X60X590 1AAA „ 1U . A 

Therefore, p, = -— — = 100.071b. persq. in. Ans. 

,ri 520 X 10 ^ h 

(549) T = 127° + 460° = 587°. Using formula 60, 
p V = . 37052 T, or V = - 37Q5 ^ x 587 ^ 8 . 055 cu. ft. Ans. 

(55G) T= 100° + 460° = 560°. 

Substituting in formula 61,/ F = .37052 W T, or 

Tjr . 37052 W T . 37052 X . 5 X 560 Q „ OK (4 _ . 
V= = ^ = 3.735 cu. ft. Ans. 

144 
(551) Use formula 64. PV=(^+^\t. 

T = 110° + 460° = 570°; T x = 100° + 460° = 560°; T % = 
130° + 460° = 590°. 

'90 X 40 , 80 X 57 > 



( ] 



1570 



Therefore, V= x 50Q 19n 59 ° ' = 67.248 cu. ft. 

1Z ° Ans. 

(552) The pressure exerted by squeezing the bulb may 
be found from formula 53, in which / is 14.7, v, the orig- 
inal volume = 20 cu. in., and v x , the new volume, = 5 cu. in. 

p v 14 7 X 20 

p —- — = — '- — — = 58.8 lb. The pressure due to the 

v x 5 

atmosphere must be deducted, since there is an equal pres- 
sure on the outside which balances it. 58.8 — 14.7=44.1 
lb. per sq. in. = pressure due to squeezing the bulb. 
3 2 X .7854= 7.0686 sq. in. = area of bottom. 7-0686 X 44.1 = 
311.725 lb. 7.0686 X .434= 3.068 lb. = pressure due to 
weight of water. 311.725 -f 3.068 = 314.793 lb. Ans. 

(553) Use formula 58. 

z>/400 + /,\ ,/4G0+115\ ln , . 
v.— ( t ,, ' )= 4( , „ ,„ )= 4.6 cu. ft. Ans. 

\460 + / / \400 4- 40 / 



STRENGTH OF MATERIALS 

(QUESTIONS 554-613.) 



(554) See Arts. 1094, 1097, and 1096. 

(555) See Arts. 1102, 1103, 1 1 lO, and 1 112, e 

(556) See Art. 1105. 

(557) Use formula 67. 

zr P l 1_ r Pl 

h = — r— ; therefore, e = — r-~r. 
A e A E 

A =.7854 X 2 2 ; /=10 X 12; P= 40 X 2,000; £=25,000,000 

m , , 40X2,000X10X12 1ftnW , 

Therefore, g = ^ ^ x ^^ = .12223-. An. 

(558) Using formula 67, 



' ^ ,-,7864 x (*)• : : .oo9 = 29 ' 708 ' 853 - 2 1K P er f^ in 

w Ans, 

(559) Using formula 67, 

u PI n AeE 1| x 2 X .006 X 15,000,000 

E = A-e> OTP =-7- t =- 9^12 ^ 

2,500 lb. Ans. 

(560) By formula 67, 

- PI 7 AeE .7854 X3 2 X. 05 X 1,500,000 __„ __,, 
^ = Z7'° r/ = ^- = ~ " 2,000 - = 265 07, 

Ans. 

(561) Using a factor of safety of 4 (see Table 24), 
formula 65 becomes 

For notice of the copyright, see page immediately following' the title page. 



216 STRENGTH OF MATERIALS. 



n A S 1 . 4P 4 X 6 X 2,000 QfVO „ „ 

P= — — i, or ^ = - r -= _ ' = .8727272 sq. m. 

4 vS oo,000 



, 4 /^4 .778727272 

{ 562) From Table 19, the weight of a piece of cast iron 
I" square and 1 ft. long is 3.125 lb.; hence, each foot of 
length of the bar makes a load of 3.125 lb. per sq. in. The 
breaking load — that is, the ultimate tensile strength — is 
20,000 lb. per sq. in. Hence, the length required to break 

the bar is 2 Q ' Q °° = 6,400 ft. Ans. 

d. 1/&0 

(563) Let t — the thickness of the bolt head; 

d = diameter of bolt. 
Area subject to shear = iz d t. 

Area subjected to tension = --d" 1 . 

S x = 55,000. 5 3 = 50,000. 
Then, in order that the bolt shall be equally strong in both 
tension and shear, - d t S 3 = — - d" 2 S v 

or , .... *d ' S t _ dS x _ j X 55,000 _ 

° r ' - i-dS, ~ IS, - 4 X 50,000 - - 20C ' AnS ' 

(564) Using a factor of safety of 15 for brick, formula 
65 gives 

p=4A 

15 ' 
A = (2| X 3J) sq. ft. = 30 X 42 = 1,260 sq. in. ; 5 Q = 2,500. 

Therefore, P = 1 » 260 X 2 »° 00 = 210,000 lb. = 105 tons. Ans. 
xo 

(565) The horizontal component of the force Pis P cos 
30° = 3,500 X .866 = 3,031 lb. The area A is 4 a, the ulti- 
mate shearing strength, 5 3 , 600 lb. , and the factor of safety, 8. 



STRENGTH OF MATERIALS. 



217 



Hence, from formula 65, 

4^S„ aS n 2P 2X3,031 



P- AS * 






8 8 2 ' " S 

(566) See Art. 1124. 



600 



= 10.1". Ans. 




Scale of forces 1=1600 lb. 
Scale of distance 1=32' 

Fig. 46. 

(567) Using formula 68, with the factor of safety of 4, 



pd — 



2/5 x _ t_S, 
4~ ; 2 



\ or t 



ipd __ % x 120 X 48 
5, 55,000 



218 STRENGTH OF MATERIALS. 

Since 40$ of the plate is removed by the rivet holes, 60$ 
remains, and the actual thickness required is 
t 2 X 120 X 48 



.60 .60X55,000 



.349". Ans. 



(568) Using a factor of safety of 6, in formula 68, 

, 2/5 tS t 

**=— = -*■' 

__ . Zpd 3X6X200 

Hence, 1 = -^ = ^^ = .18 . Ans. 

(569) Using formula 71, with a factor of safety of 10, 

P ~ WTd = 960,000-^. 



.-is/ //</ «-i/130 X 12 X 12 X 3 

HenCG ' ' = ^ 96^00-0 = V 96^000 = ^ ' 

Ans. 

(570) From formula 70, 

St pr 2,000 X | _ 4,000 „ 

' ~ T+7' ° r ' - 5=7 " 2,800 - 2,000 ~ "800" ~ ° " AnS ' 

(571) See Fig. 46. (a) Upon the load line, the loads 
0-1, 1-2, and 2-3 are laid off equal, respectively, to 
F 2 , F 3 , and F t ; the pole P is chosen, and the rays drawn in 
the usual manner; the pole distance H = 2,000 lb. The 
equilibrium polygon is constructed by drawing a c, c d, d e, 
and e f parallel to P 0, Pi, P 2, and PS, respectively, and 
finally drawing the closing line/" a to the starting point a. 
P m is drawn parallel to the latter line, dividing the load 
line into the reactions m = R j9 and 3 in = R 2 . The shear 
axis m 11 is drawn through m, and the shear diagram 
h I . . . . s 11 in is constructed in the usual manner. To 
the scale of forces in = 1,440 lb., and 3 in = 2,160 lb. To 
the scale of distances the maximum vertical intercept y = 
d'd=31.% ft., which, multiplied by H,— 31.2 X 2,000 = 
62,400 ft.-lb. = 748,800 in. -lb. Ans. 

(/;) The shear at a point 30 ft. from the left support = 
0m = 1,440 lb. Ans. 

(c) The maximum shear = ns' == — 2,160 lb. Ans. 



STRENGTH OF MATERIALS. 



219 



(572) See Fig. 47. Draw the force polygon 0-1-2-3-^-5-0 

in the usual manner, 0-1 being equal to and parallel to 
F i9 1-2 equal to and parallel to F 91 etc. 0-5 is the resultant. 

Scale of forces 1=40 lb* 
Scale of distance 1=2" 



m<-^ 




Fig. 47. 



Choose the pole P, and draw the rays P 0, Pi, P 2, etc. 
Choose any point, a on F v and draw through it a line 
parallel to the ray P 1. From the intersection b of this line 
with F 2 , draw a line parallel to P 2; from the intersection 
c of the latter line with F % produced, draw a parallel to 
P 3, intersecting F^ produced in d. Finally, through d f 
draw a line parallel to P If, intersecting F b produced in e. 
Now, through a draw a line parallel to P 0, and through 
e a line parallel to P 5; their intersection/" is a point on the 
resultant. Through / draw the resultant R parallel to 
0-5. It will be found by measurement that R = 65 lb., that 
it makes an angle of 22-§-° with m n, and intersects it at a 
distance of lY from the point of intersection of F x and /// n. 

(573) See Fig. 48. The construction is entirely similar 
to those given in the text. 0-1, 1-2, and 2-3 are laid or! to 
represent F t F 2 , and F 3 ; the pole P is chosen and the rays 



220 



STRENGTH OF MATERIALS. 



drawn. Parallel to the rays are drawn the lines of the 
equilibrium polygon a b c d g a. The closing line g a is 
found to be parallel to P 1. Consequently, 0-1 is the left 
reaction and 1-3 the right reaction, the former being 6 tons 
Ft T* R 

e\Tons. 2iTons. liTon. 



±e 




Scale of forces 1=5 tons. n * 

Scale of distance 1-5' 

Fig. 48. 

and the latter 3 tons. The shear diagram is drawn in the 
usual manner; it has the peculiarity of being zero between 
F x and F t . 

(574) The maximum moment occurs when the shear 
line crosses the shear axis. In the present case the shear 
line and shear axis coincide with s t, between P l and F u ; 
hence, the bending moment is the same (and maximum) at 
F x and F 2 , and at all points between. This is seen to be true 
from the diagram, since k h and b c are parallel. Ans. 

(b) By measurement, the moment is found to be 24 X 12 = 
288 inch-tons. Ans. 

(r) 288 X 2,000 = 576,000 inch-pounds. Ans. 



STRENGTH OF MATERIALS. 
(575) See Arts. 1 133 to 1 137. 



221 



(576) See Fig. 49. The force polygon 0-1-2-3-^-0 is 
drawn as in Fig. 47, 0-4. being the resultant. The equilibrium 
polygon a b c d sr a 



the 



Scale of forces 1=50 lb. 
Scale of distance 1=2" 



a 
is then drawn, 
point g lying on the 
resultant. The re- 
sultant R is drawn 
through g, parallel to 
and equal to 0-4- A 
line is drawn through 
C, parallel to R. 
Through g the lines 
ge and gf are drawn 
parallel, respectively, 
to PO and P^ and 
intersecting the par- 
allel to R, through C 
in e and f ; then, ef 
is the intercept, and 
Pu, perpendicular to 
0-4, is the pole dis- 
tance. Pu = 33 lb. ; 
ef=1.32". Hence, 
the resultant mo- 
ment is 33 X 1.32 = 43.6 in. -lb. Ans. 




,/./„ 



(577) The maximum bending moment, M= JV-^ (see 



Fig. 6 of table of Bending Moments) = 4x 2,000 X 



/ 
14 x 8 

22 



40,727 T 3 T ft. -lb. =488,727 in. -lb. Then, according to for- 
mula 74, 



fc 



= 488,727. 



I _ 488,727/ _ 488,727 X 8 
c ~ S\ 9,000 



434.424. 



222 



STRENGTH OF MATERIALS. 



' c \d 



~b d*, and, according to the conditions 
6 



of the problem, b — — d. 



Therefore, - = \b d 2 = ^-d* = 434.424. 
c 6 12 



d 3 = 5,213.088 
d = 17i". J 
b = 81". J 



Ans. 



(578) The beam, with the moment and shear diagrams, 
is shown in Fig. 50. On the line, through the left reaction, 

1.3 i Tons. 




Fig. 50. 



are laid off the loads in order. Thus, 0-1 = 40 X 8 = 320 lb., 
is the uniform load between the left support and P\ ; 1-2 is 



STRENGTH OF MATERIALS. 223 

F x = 2,000 lb. ; 2-3 =40 X 12 = 480 lb., is the uniform load 
between F x and F 2 ; 3-lf. = 2,000 X 1.3 = 2, GOO lb., is F v and 
jf.-5 — 40 X 10 = 400 lb., is the uniform load between F 2 and 
the right support. The pole P is chosen and the rays drawn. 
Since the uniform load is very small compared with F x and F 2 , 
it will be sufficiently accurate to consider the three portions 
of it concentrated at their respective centers of gravity 
x, y, and z. Drawing the equilibrium polygon parallel to 
the rays, we obtain the moment diagram a h b k c I d a. 
From P, drawing P m parallel to the closing line a d, we 
obtain the reactions m and m 5 equal, respectively, to 2,930 
and 2,870 lb. Ans. The shear axis m x, and the shear dia- 
gram r s t u v n m, are drawn in the usual manner. The 
greatest shear is m, 2,930 lb. The shear line cuts the shear 
axis under F r Hence, the maximum moment is under F s . By 
measurement, <?ris64 // , andPx is 5,000 lb. ; hence, the maxi- 
mum bending moment is 64 X 5,000 = 320,000 in. -lb. Ans. 

(579) From the table of Bending Moments, the great- 

wP 
est bending moment of such a beam is —5—, or, in this case, 

8 

w X 240 2 



By formula 74, 



n/r _ w X 240 2 _ SJ _ 45,000 280 

ivi — — ■ — — -- — - ■ x 



8 fc 4 12 -=- 2 

rru f 45,000 X 280 X 8 ^ no „ . -, , 

Therefore, w — —?■ — = 72.92 lb. per inch of 

240 X 240 X 4 X 6 v 

length = 72.92 X 12 = 875 lb. per foot of length. Ans. 

(580) From the table of Bending Moments, the maxi- 
mum bending moment is 

4 4 

From formula 74, 

I=~(d i -d l i ) = 5Q.U5;c=^d= ( ^-=3^-, 5 =38,000;/=6 



224 STRENGTH OF MATERIALS, 

38,000 X 56.945 



Hence, 24 W- 



6 X 3.25 



„. 38,000 X 56.945 t ^^ u A 

W= — ^ — - — = 4, 624 lb. Ans. 

24 X 6 X 3.25 



(581) ( a ) From the table of Bending Moments, 
n/r wl" WX192 2 

M= ^= — ' 

From formula 74, 

nf w X 192 2 SJ 
M= 8 = /T 

S 4 = 7,200;/=8; /= ±bd* = ^J°; c = \d= 5. 

mu w X 192 2 7,200 2,000 
Then, = - 1 — — x 



8 8 12 X 5 

7,200 X 2,000 X 8 
W ~ 8 X 12 X 5 X 192 X 192 " 
6.51 lb. per in. = 6.51 X 12 = 78.12 lb. per ft. Ans. 

,. w 1 7 „ 10 X 2 3 80 

w X 192 2 _ 7,200 80 

^ " — ?; " X 



12 X 1" 

7,200 X 80 X 8 . Q „ 

zc/ = = 1.3 lb. per in. 

8 X 12 X 192 X 192 y 

= 1.3 X 12 = 15.6 lb. per ft. Ans. 

(582) {a) From the table of Bending Moments, the 

deflection of a beam uniformly loaded is — — ^ . . In 

J 384 El 

Example 579, 1^=874x20=17,480 lb.; / = 240", £ = 

25,000,000, and /= 280. 

xj a a *■ 5 X 17,480 X 240 3 IJS . . 

Hence, deflection s — — — _ ^^ ^^ — — = . 4o in. Ans. 

384 x2o, 000, 000 X 280 

(b) From the table of Bending Moments, s = — p . 



STRENGTH OF MATERIALS. 225 

In Example 580, W= 4,624 lb. ; / = 96 in. ; E = 15,000,000, 

and 7=56.945. 

„ 4,624 X 96 3 .„ . A 

Hence, s = — „ „ ' ^ nnn ————• = . 1 , nearly. Ans. 

48 X 15,000,000 X 56.945 ' y 

(c) s = ^Jff- In Example 581 (a), 1^=78.12X16; 

/=192; £=1,500,000, and 1= M??. 

12 

„ 5 X 78.12 X 16 X 192 3 . ( ,.„ . 

HenC6 -^ 384 X 1,500,000 XHI^ - 461 ' ^ 

(583) Area of piston = ^tt d* = 5- tt x 14 2 . 

W= pressure on piston = — rr x 14 2 X 80. 

From the table of Bending Moments, the maximum 
bending moment for a cantilever uniformly loaded is 

wP Wl ^ X 14 2 X80X4 5 4 / c , y „■ 

"Y" = ~2~ = ~ 2 = / T formula ?4. 

„ i^Xl4 2 X80x4 4,500;r^ 3 

Hence, - — ~- 



2 32 

,, 14 2 X 80 X 4 X 32 

° rd = 4X2X4,500 - 55 - 75 ' 

^=^55?75 = 3.82''. Ans. 

(584) Substituting in formula 76, 5 2 = 90,000; A = 

6 2 X .7854; /=6; /= 14x12 = 168; £-=5,000; ^=^-X 

6\ we obtain 

W- S > A - 90.000 X 6 8 X .7854 _ 

^n ■ ^Wn i O'X. 7854X168' v-l~0.8'^b- 

o4 

(585) For timber, 5 3 = 8,000 and /=8; hence, -y = 

-1— - = 1,000. 



226 STRENGTH OF MATERIALS. 

Substituting in formula 65, 

P= A -^=1,000 A. 
. P 7 X 2,000 . . , 

= Fooo = i QQQ = u sq - m -» necessar y area of a 

short column to support the given load. Since the column 
is quite long, assume it to be 6" square. Then, A — 36. 

and/ =iV=S= 108 - 

Formula 76 gives 

I = 30 X 12 = 360, and g = 3,000. 
S, _ 14,000 / 36 X 360' X _ 

/- 36 i 1 + 3,000Xl08J-°' JJ0 ' nearIy ' 
Since this value is much too large, the column must be 
made larger. Trying 9" square, A = 81, /= 5-iGf. 
Then ^ _ 14000 L . 81 X 360 X 3C0 X _ ow 
Then ' 7~~^^l + T"000x546f>|- 1 ' 2,J - 

This value of -~ is much nearer the required value, 1,000. 

Trying 10" square, A = 100, 1= 1M22. = 833J,' 

12 

S, 14,000 / 100x360x360\ _„„ , 

/ = ^00- l 1 + 3,000 X 833^ ) = 866 ' "^ 

Since this value of -~ is less than 1,000, the column is a 

little too large; hence, it is between 9 and 10 inches square. 

9f" will give 997.4 lb. as the value of -^; hence, the column 

should be 9§" square. 

This problem may be more readily solved by formula 
77, which gives 

_ a / 7 X 2000 x 8 / 7 x 2000 X 8 / 7 x 2000 x 8 13 X 360* \ _ 

C ~ * 2 x 800 + t 8000 V 4 X 8000 h 3000 J ~ 

4/7+ ^^TT(3.5 + 518.4) = 4/92.479 = 9.61" = 9f ", nearly. 



STRENGTH OF MATERIALS. 227 

(586) Here W = 21,000; /= 10 ; 5 2 = 150,000; g = 
6,250; / = 7.5 X 12 = 90". For using formula 78, we have 

.3183 Wf _ .3183 X 21,000 X 10 



5 2 150,000 

16 / 2 16X8100 



= .4456. 



g 6250 

Therefore, 



= 20.7360. 



d—\. 4142 4/.4456 + I/. 4456 (.4456 -f- 20. 7360) = 
1. 4142 j/. 4456 + 3.0722 = 2.65", or say 2f". 

(587) For this case, ^4 = 3.1416 sq. in.; /=4xl2 = 
48"; S 2 = 55,000; /= 10; / = .7854; ^ = 20,250. 
Substituting these values in formula 76, 

„. SA 55,000 X 3.1416 - 

W — 



/(.+# ) k 



3.1416 X 48' 



20,250 X .7854/ 
5,500 X 3.1416 
1.4551 
Steam pressure = 60 lb. per sq. in. 

Then, area of piston = .7854 d* = - = ' L46fil x 60 ■ 

Hence > d% = .isdx?AMi™eo = 252 ' nearly ' 



and d = 4/252 = 15 J-", nearly. Ans. 

(588) (a) The strength of a beam varies directly as the 
width and square of the depth and inversely as the length. 
Hence, the ratio between the loads is 

6X 8 2 . 4X 12 2 .. _ 1 ■ . 

-Jq- : 16 = 16 : 15, or 1^. Ans. 

(b) The deflections vary directly as the cube of the 
lengths, and inversely as the breadths and cubes of the 
depths. 

Hence, the ratio between the deflections is 
10 3 16 3 



6 X 8 3 4 X 12 a 
a. a. IV.— ij, 



= .549. Ans. 



228 STRENGTH OF MATERIALS. 

(589) Substituting the value of c lt from Table 27 in 
formula 80, we obtain 

(a) d=c/^=4. 92 X^— = 3. 739'. Ans. 

(590) Using formula SO, 



4,000 



d= 5. 59 j/^ = 14.06". 

Since this result is greater than 13.6", formula 81 must 
be used, in which 



rf = * 1 f#=8.3f^°=^a»». Ans. 



(591) From formula 80, 

77 

N 



s/77 d A N 

d=c i Y- w ,orH=—r r -. ^ = 4.11. (Table 27:) 



Hence, H = ^-^ = 71. 775 H. P. Ans. 
4. II 4 

(592) Using formula 83, 

^=^A^ ^ 4 ~^A = ,0212X100( (7 ^ 7 4 X ~ 54 ) = 717.7H.P. Ans 
(.0212 is the value of q 1 from Table 28.) 

(593) (a) Using formula 84, 

P= 100 C 2 = 100 X 8 2 = 6,400 lb. Ans. 
(b) Using formula 85, 

C=i/l0= 3.162". 

d=\ C= 1.054". Ans. 
o 

(r) Using formula 86, 

/> — 1 000 <f a - C a — ^ _ 6 t X 2,000 _ 

J ' uuuc >c -1,000" 1,000 - ld *- 
C= 4/131 = 3.651". Ans. 



STRENGTH OF MATERIALS. 229 

(594) (a) Using formula 87, 

P= 12,000 d 2 = 12,000 X (~) = 9,187.5 lb. Ans. 
(b) Formula 88 gives P= 18,000 d\ 



Therefore, d= /^ = f^M = V\ =-667". An, 

(595) The deflection is, by formula 75, 

wr i wr 

s = a r- t = TT^r t- T , the coefficient being found from the 
£/ 192 EI 

table of Bending Moments. 

Transposing, ^=^^ ; /= 120; E- 30,000,000; / = 

.7854; j = |. 

Then, ^ = 192 X 30 -° QQ ; Q 9 Q n ? X - 7854 = 327.25 lb. An, 
o X 120 

(596) (a) The maximum bending moment is, accord- 

* 4.1, l U1 rn r ivT 4- ^ 6 >°00 X 60 

mg to the table of Bending Moments, — — = — - = 

90,000 inch-pounds. 

By formula 74, 

S I 

M =90,000 = -^ -. 
/ c 

I 64 7rd* 



S i = 120,000; /=10. 7 =:-- 
Hence, 90,000 = 



^ id 32 
120,000 tt^ 3 



10 32 



^ 90,000 X 10 _Xj2 _ _ 

° r ^ ~ y 120,000 X 3.1416 ~ 4,M4: ~ 44 ' nearly * 

(I?) Using formula 80, 

rf = '. r -y = 4. 7 f g = 4g', nearly. Ans. 



230 



STRENGTH OF MATERIALS. 



(597) (a) The graphic solution is shown in Fig. 51. 
On the vertical through the support 0-1 is laid off equal to 
the uniform load between the support and F } ; 1-2 is laid off 



350 }f lb. 
5 t 6 IL _A 




Scale of forces 1^800 lb. 
Scale of distance 1—4* 



Fig. 51. 



to represent F Jt 2-3 represents to the same scale the re- 
mainder of the uniform load, and 3 m represents F r The 
pole P is chosen and the rays drawn. The polygon a be efh 
is then drawn, the sides being parallel, respectively, to the 
corresponding rays. If the uniform load between F y and F 9 
be considered as concentrated at its center of gravity, the 
polygon will follow the broken line c e f. It will be better 
in this case to divide the uniform load into several parts, 
2-4, lf-5, 5-6, etc., thus obtaining the line of the polygon 



STRENGTH OP MATERIALS. 231 

c d f. To draw the shear diagram, project the point 1 
across the vertical through F v and draw O s. Next project 
the point 2 across to /, and 3 across to «, and draw t u. 
s t it 11 in is the shear diagram. The maximum moment 
is seen to be at the support, and is equal to a Ji X P m. To 
the scale of distances, a // = 58.8 in., while Pm = H — 
1,400 lb. to the scale of forces. Hence, the maximum bend- 
ing moment is 58.8 X 1,400 = 82,320 in.-lb. Ans. 
(b) From formula 74, 

M=^- = 82,320. 5 4 = 12,500; /= 8. 

_, , / 82,320 X 8 KO _ Q 
Therefore, -= L__ = 5 2.68. 

_ 4 / lfbd % bd* A , __ , . 2d 
But > 7 = HFT = T' and d = 2 ^' or * = T" 

Hence, ~=~- = ^ = 52.68. </ 3 = 52.68 X .15 = 790.2. 
c b 15 

2^ 



dT = f 790.2 = 9.245". £ = — = 3.7", nearly. Ans. 
(598) Referring to the table of Moments of Inertia, 
(P d* - b x d?y - Ibdb, d^d- dy _ 
Vl{bd-b^) 
[8 X 10 2 - 6 X (8j) 2 ] 2 -4X8X10X6X81(10- 8£)' 
12 (8 X 10 - 6 X 8i) 

280.466. 
d . b. d. / </ — <^ 

10 , 6 X 81 



2 ' 2 



^X / d-d \ _ 
+ 2 \bd-b x dj~ 



(#) From the table of Bending Moments, the maximum 
bending moment is — — . 

S t = 120,000; /= 7; /=35 X 12= 420 in. 

Wl S I 
Using formula 74, M= —-— = -4— t or 

4 J c 

T „ 4 5 4 7 4X120,000X280.466 w „,■. „ 

fF= , J = , rt „ „ ^-r— = 7. 246 lb. Ans. 

Ifc 420 X 7 X 6.319 



232 



STRENGTH OF MATERIALS. 



(b) In this case f = 5, and the maximum bending 

w r 

moment is —5—. Hence, from formula 74, 



M 



-, or w 



Therefore, W = wl 



A' ' 2 A' 

8 5/ 8 X 120,000 X 280.46G 



Ifc 



•420 X 5 X 6,319 



20,290 lb. Ans. 

(599) (a) According to formula 72, 



f= A r 2 , or r = \ 



w? 



4/3 = 1.732. Ans. 




Scale of forces 1=960 lb. 
Scale of distance 1-8' 

Fig. 52. 



(b) From the table of Moments of Inertia, I=—5d* = 



72 ; A = bd=2±. Dividing, 
36; d = 6" and b = ±" . Ans. 



bd 



g, or 1^ = 3. «/■ = 



STRENGTH OF MATERIALS. 233 

/ Vd 1 

4 

(600) Using formula 69, pd—US, we have t = £—r. 

4 o 

Using a factor of safety of 6, 

, 4/5 6/</ 6X100X8 

^ = _ f . or/ = ^ = ____ = .06. Ans. 

(601) The graphic solution is shown in Fig. 52. The 
uniform load is divided into 14 equal parts, and lines drawn 
through the center of gravity of each part. These loads 
are laid off on the line through the left reaction, the pole P 
chosen, and the rays drawn. The polygon be d e f a is then 
drawn in the usual manner. The shear diagram is drawn 
as shown. The maximum shear is either t 7 or r v — 540 lb. 
The maximum moment is shown by the polygon to be at 
/Y vertically above the point u, where the shear line crosses 
the shear axis. The pole distance P 7 is 1,440 lb. to the 
scale of forces, and the intercept /Y is 14 inches to the scale 
of distances. Hence, the bending moment is 20,160 in. -lb. 

(602) From formula 74, 

S I 

M= y- = 20,160. 5 4 = 9,000; /= 8. 

But, — = 12 , , = — b d* for a rectangle. 
' c \d 6 s 

Hence, \bd* = 17.92, or bd 2 = 107.52. 

Any number of beams will fulfil this condition. 

Assuming d = 6", b = — -^ — = 3", nearly. 

Assuming;/ =5', b = 



25 

(603) Using the factor of safety of 10, in formula 71 
9,600,000 ^ = 960 000 X .2'- = ^ % 

F 10 Id 108 X 2.5 



234 STRENGTH OF MATERIALS. 

(604) Using formula 87, 
P= 12,000a 72 . or d 



i o (\c\r\ 



5 X 2,000 



913". Ans. 



12,000 r 12,000 

(605) The radius r of the gear-wheel is 24". Using 
formula 80, d = c \/P~r = . 297 ^350 X 24 = 2. 84". Ans. 

(606) Area of cylinder = .7854 X 12 2 = 113.1 sq. in. 

Total pressure on the head = 113.1 X 90 = 10,179 lb. 

10 179 
Pressures on each bolt = — ?—— = 1,017.9 lb. 

Using formula 65, 

P 1 017 9 
P—A S, or A = -~ = ! '' = .5089 sq. in., area of bolt. 



Diameter of bolt 



= /■■ 



5089 



= .8", nearly. Ans. 



.7854 

(607) (a) The graphic solution is clearly shown in Fig. 
53. On the vertical through F if the equal loads P 1 and F a 

Fi F, 

1 Ton. 1 Ton. 




Scale of forces f=2000 lb. 
Scale of distance 1=6* 

s Fig. 53. 



are laid off to scale, 0-1 representing F t and 1-2 representing 



STRENGTH OF MATERIALS. 235 

F r Choose the pole P, and draw the rays P 0, P 1, P 2. 
Draw a b between the left support and F x parallel to P 0; 
b c between F l and F 2 parallel to P 1, and c d parallel to 
P 2 y between F 2 and the right support. Through P draw a 
line parallel to the closing line a d. 0-1 — 1-2 \ hence, the 
reactions of the supports are equal, and are each equal to 1 
ton. The shear between the left reaction and F x is nega- 
tive, and equal to F x = 1 ton. Between the left and the 
right support it is 0, and between the latter and F a it is posi- 
tive and equal to 1 ton. The bending moment is constant 
and a maximum between the supports. To the scale of 
forces P 1 = 2 tons = 4,000 lb., and to the scale of distances 
a f— 30 in. Hence, the maximum bending is 4,000 X 30 
= 120,000 in. -lb. Ans. 

(b) Using formula 74, 

S 1 T 
M= -J - = 120,000. 5 4 = 38,000; /= 6. 
f c 4 ■ 

_, / 120,000X6 360 1ft . 

Th6n ' 7 = -3^000- = 19" = 19 ' nearl y- 

itd* 



_, ■ / 64 * d< 
But, - = 



c d ' 32 



tt *d % • 32X19 

Hence,— = 19, or ^.-^^ 



= ^ 



3 ™ = 5 - 784 - A - 



(608) Since the deflections are directly as the cubes of 
the lengths, and inversely as the breadths and the cubes 
of the depths, their ratio in this case is 

18 3 . 12 3 27.9 

2 X 6 3 : 3 X 8 3 ' ° r 2" : 8 ~ 

That is, the first beam deflects 12 times as much as the 
second. Hence, the required deflection of the second beam 
is .3 -v- 12 = .025". Ans. 



236 STRENGTH OF MATERIALS. 

(609) The key has a shearing stress exerted on two 
sections; hence, each section must withstand a stress of 

20 > 000 1A AAA A 

— L. = 10,000 pounds. 

2 

Using formula 65, with a factor of safety of 10, 

n AS^ . 10 P 10X10,000 n 

/>= — % or A=-^= 5() ^ = 2 sq. in. 

Let b = width of key; 
/ = thickness. 

Then, b t = A — 2 sq. in. But, from the conditions of the 
problem, 



1 

4 

2.828 



Hence, b t = ±-b 2 = 2; £ a = 8; £ = 2.828' 



/ = 



Ans. 



(61 0) From formula 75, the deflection s = a—^y, 

and, from the table of Bending Moments, the coefficient a for 

the beam in question is-—. 

48 

W= 30 tons = 00,000 lb. ; / = 54 inches; E = 30,000,000; 



/ = 



64 



„ IX GO, 000 X 54 3 nneA . 

Hence, s = '■ =.0064 m. 

48 X 30,000,000 X ^ X Ans. 

(611) (a) The circumference of a 7-strand rope is 3 
times the diameter; hence, C= 1J X 3= 3f". 

Using formula 86, /> = 1,000 C* = 1,000 X (3|) 9 = 

14,062.5 1b. Ans. 

(£) Using formula 84, 



/>=100 C -,orC = l/^ = /li^000 =5 .92'. Ans. 



STRENGTH OF MATERIALS. 



237 



(61 2) Using formula 70, 



/ = 






120,000 



= 12,000 lb. Ans. 



(613) The construction of the diagram of bending 




m2 



Scale of forces 1=1600 lb. 
Scale of distance 1=32' 

Fig. 54. 



238 STRENGTH OF MATERIALS. 

moments and shear diagram is clearly shown in Fig. 54. It 
is so nearly like that of Fig. 46 that a detailed description 
is unnecessary. It will be noticed that between k and k' 
the shear is zero, and that since the reactions are equal the 
shear at either support = % of the load = 2,400 lb. The 
greatest intercept is c c' = d d' = 30 ft. The pole distance 
H — 2,400 lb. Hence, the bending moment = 2,400 X 30 = 
72,000 ft. -lb. = 72,000 X 12 = 804,000 in. -lb. 



SURVEYING. 

(QUESTIONS 614-705.) 



(614) Let ^r = number of degrees in angle C ; then, 
2x = angle A and 3x = angle B. The sum of A, B, and C 
isx + 2x+3x=Qx = 180°, or x = 30° = C ; &tr = 60° = ^, 
and 3-r = 90° = B. Ans. 

(615) Let x = number of degrees in one of the equal 
angles; then, %x = their sum, and %x X 2 = 4,r = the greater 
angle. 2-r -f- 4:X = 6.r = sum of the three angles = 180°; 
hence, # — 30°, and the 



greater angle = 30 c 
120°. Ans. 



X 4 = 




(616) AB in Fig. 55 
is the given diagonal 3.5 in. 
3. 5 2 = 1 2.25 -T- 2 = 6.125 in. 
|/6.125 = 2.475 in. = side of 
the required square. From 
A and B as centers with 
radii equal to 2.475 in., de- 
scribe arcs intersecting at 
C and D. Connect the ex- 
tremities A and B with the 
points C and D by straight 
lines. The figure A C B D is the required square. 

(617) Let A B, Fig. 56, be the given shorter side of the 
rectangle, 1.5 in. in length. At A erect an indefinite per- 
pendicular A C to the line A B. Then, from B as a center 
with a radius of 3 in. describe an arc intersecting the per- 
pendicular A C in the point D. This will give us two 

For notice of the copyright, see page immediately following the title page. 



240 



SURVEYING. 



A 




\D 








\ 


t 


P^ 






*«j 








B 




G 






Fig. 56. 



adjacent sides of the required rectangle. At B erect an 
indefinite perpendicular BE to A B, and at D erect an in- 
definite perpendicular D F to AD. These perpendiculars 

will intersect at G, and the 
resulting figure ABGD will 
be the required rectangle. Its 
area is the product of the 
length A D by the width A B. 
A D* = B~D* - ~AB\ BD 2 = 
--■ E 9 in. ; A B* = 2.25 in. ; hence, 
AD* = 9 in. - 2.25 in. = 6.75 
in. 4/6.75 = 2.598 in. = side 
A D. 2,598 in. X 1.5 = 3.897 sq. in., the area of the required 
rectangle. 

(618) (See Fig. 57.) With 
the two given points as cen- 
ters, and a radius equal to 
3. 5*4- 2 = 1.75*= IF, describe 

short arcs intersecting each 
other. With the same radius 
and with the point of intersec- 
tion as a center, describe a 
circle; it will pass through 
the two given points. 

B in Fig. 58 is the given line, A the given 
point, A C and C B — A B. 
The angles A, B, and C are 
each equal to 60°. From B 
and C as centers with equal 
radii, describe arcs intersect- 
ing at D. The line A D bi- 
sects the angle A ; hence. 
Fig. 58. angle BAD — 30°. 





(620) Let A B in Fig. 59 be one of the given lines, 
whose length is 2 in., and let A C, the other line, meet A B 
at A, forming an angle of 30°, From A and B as centers, 



SURVEYING. 



241 



with radii equal to A B, describe arcs intersecting at D. 
Join A D and B D. The triangle A B D is equilateral.; 
hence ; each of its angles, as A, contains 60°. From D as a 




Fig. 59. 



center, with a radius A D, describe the arc A B. 
A C is tangent to this arc at the point A. 



The line 



(621) A B in Fig. 60 is the given line, C the given 
point, C D = C E. 
From D and E as cen- 
ters with the same 
radii, describe arcs in- 
tersecting at/ 7 . 
Through F draw C G. 
Lay off C G and C B 
equal to each other, 
and from B and G as 
centers with equal 
radii describe arcs intersecting at H. 
angled C 77=45°. 




Fig. 60. 



Draw C H. The 



2-12 



SURVEYING. 



(622) 1st. A B in Fig. 61 is the given line 3 in. long. 
At A and B erect perpendiculars A C and B D each 3 in. 

long. Join A D and B C. 



CK 



A D 



/ 



\e/ 



/ 




90 



*S\ 



These lines will intersect at 
some point E. The angles 
E A B and E B A are each 45°, 
and the sides A E and E B 
must be equal, and the angle 
A E £= 90°. 

2d. A B in Fig. 62 is the 
given line and c its middle 
point. On A B describe the 
semicircle A E B. At c 

erect a perpendicular to A B, cutting the arc A E B in E. 

Join A E and E B. The angle 

A EB is 90°. 






Fig. 61. 



(623) See 
and Fig. 237. 



Art. 1181 



(624) See Art. 1187. 

(625) (See Art. 1195 
and Fig. 245.) Draw the line 
A B, Fig. 63, 5 in. long, the length of the given side. At 
A draw the indefinite line A C, making the angle B A C 





Fig. 63. 



equal to the given angle of 30°. On A C lay off A D 1.5 
in. long, the given difference between the other two sides of 



SURVEYING. 



243 



the triangle. Join the points B and D by a straight line, 
and at its middle point E erect a perpendicular, cutting the 
line A C in the point F. Join B F. The triangle A B F is 
the required triangle. 

(626) See Art. 1198. 

(627) See Art. 1200. 

(628) See Art. 1201. 

(629) See Art. 1204. 

(630) See Art. 1205. 

(631) See Arts. 1205 and 1206. 

(632) See Art. 1206. 

(633) See Art. 1 204. 

(634) See Art. 1207. 

(635) See Art. 1207. 

(636) See Arts. 1209 and 1213. 

(637) See Art. 1211. 

(638) (a) In this example the declination is east, and 



Magnetic 
Bearing. 


True 
Bearing. 


N 15° 20' E 


N 18° 35' 


E 


N 88° 50' E 


S 87° 55' 


E 


N 20° 40' W 


N 17° 25' 


W 


N 50° 20' E 


N 53° 35' 


E 



for a course whose magnetic bearing is N E or S W, the 



G. a. IV 



-15 



244 SURVEYING. 

true bearing is the sum of the magnetic bearing and the 
declination. For a course whose magnetic bearing is N W 
or S E, the true bearing is the difference between the mag- 
netic bearing and the declination. 

As the first magnetic bearing is N 15° 20' E, the true 
bearing is the sum of the magnetic bearing and the declifia- 
tion. We accordingly make the addition as follows : 

N 15° 20' E 
3° 15' 



N 18° 35' E, 



and we have N 18° 35' E as the true bearing of the first 
course. 

The second magnetic bearing is N 88° 50' E, and we add 
the declination of 3° 15' to that bearing, giving N 92° 05' E. 
This takes us past the east point to an amount equal to the 
difference between 90° and 92° 05', which is 2° 05'. This 
angle we subtract from 90°, the total number of degrees be- 
tween the south and east points, giving us S 87° 55' E for 
the true bearing of our line. A simpler method of deter- 
mining the true bearing, when the sum of the magnetic 
bearing and the declination exceeds 90°, is to subtract that 
sum from 180° ; the difference is the true bearing. Apply- 
ing this method to the above example, we have 180° — 
92° 05'= S 87° 55' E. 

The third magnetic bearing is N 20° 30' W, and the true 
bearing is the difference between that bearing and the 
declination. We accordingly deduct from the magnetic 
bearing N 20° 40' W, the declination 3° 15', which gives 
N 17° 25' W for the true bearing. 

(b) Here the declination is west, and for a course whose 
magnetic bearing is N W or S E the true bearing is the sum 
of the magnetic bearing and the declination. For a course 
whose magnetic bearing is N E or S W, the true bearing is 
the difference between the magnetic bearing and the 
declination. 



SURVEYING. 



245 



The first magnetic bearing is N 7° 20' W, and as the 
declination is west, it will be added. We, therefore, have 
for the true bearing N 7° 20' W + 5° 10' = N 12° 30' W. 



Magnetic Bear- 
ing. 


True 
Bearing. 


N 7° 20' W 


N 12° 30' 


W 


N 45° 00' E 


N 39° 50' 


E 


S 15° 20' E 


S 20° 30' 


E 


S 2° 30' W 


S 2° 40' 


E 



The next two bearings the student can readily determine 
for himself. 

The fourth magnetic bearing is S 2° 30' W, and to obtain 
the true bearing we must subtract the declination, i. e. , we 
must change the direction eastwards. A change of 2° 30' will 
bring us due soutJi ; hence, the bearing will be east of south 
to an amount equal to the difference between 2° 30' and the 
total declination 5° 10', which is 2° 40'. The true bearing 
is, therefore, S 2° 40' E. 

(639) See Art. 1216. 

(640) See Art. 1217. 

(641) See Art. 1219. 

(642) A plat of the accompanying notes is given in 
Fig. 64 to a scale of GOO ft. to the inch. The order of work 
is as follows: 

First draw a meridian N S (see Fig. 64), and then assume 
the starting point A, which call Sta. 0. Through A draw 
a meridian A B parallel to N S. Then, placing the 
center of the protractor at A, with its zero point in the 
line A B, lay off the bearing angle 10° 10' to the right of 



246 



SURVEYING. 



A B, as the bearing is 
N E. Mark the point of 
angle measurement care- 
fully, and draw a line join- 
ing it and the point A. 
This line will give the di- 
rection of the first course, 
the end of which is at 
Sta. 5 + 20, giving 520 

ft. for the length of that course. On this 
line lay off to a scale of 600 ft. to the inch the 
distance 520 ft., locating the point C, which is 
Sta. 5 -f- 20. Through £7 draw a meridian CD, and 
with the protractor lay off the bearing angle 
N 40° 50' E to the right of the meridian, mark- 
ing the point of angle measurement and joining 
it with the point C by a straight line, which will 
be the direction of the second course. The end 
of this course is Sta. 10 -f- 89, and its length is 
the difference between 1,089 and 520, which is 




$ 



Css 



rrs 




Fig. 64. 

569 ft. This distance scale off from C, locat- 
ing the point E at Sta. 10 -J- 89. In a similar 
manner plat the remaining courses given in 
the example. Write the bearing of each line 
directly upon the line, taking care that the 
bearings shall read in the same direction 
as that in which the courses are being 
run. 



SURVEYING. 247 

(643) See Art. 1231 and Figs. 261 and 262. 

(644) For first adjustment see Art. 1233. For 
second adjustment see Art. 1234 and Fig. 263, and for 
//^"readjustment see Art. 1235 and Fig. 264. 

(645) See Art. 1238 and Fig. 265. 

(646) See Art. 1239 and Fig. 266. 

(647) See Art. 1 240 and Fig. 267. 

(648) See Art. 1242 and Fig. 269. 

(649) See Art. 1242 and Fig. 269. 

(650) To the bearing at the given line, viz. , N 55° 15' E, 
we add the angle 15° 17', which is turned to the right. This 
gives for the second line a bearing of N 70° 32' E. 

(651) To the bearing of the given line, viz., N80°H' E,we 
add the angle 22° 13', which is the amount of change in the 
direction of the line. The sum is 102° 24', and the direction 
is 102° 24' to the right or east of the north point of the com- 
pass. At 90° to the right of north the direction is due east. 
Consequently, the direction of the second line must be south 
of east to an amount equal to the difference between 
102° 24' and 90°, which is 12° 24'. Subtracting this angle 
from 90°, the angle between the south and east points, we 
have 77° 36', and the direction of the second line is S 77° 36' E. 
The simplest method of determining the direction of the 
second line is to subtract 102° 24' from 180° 00'. The differ- 
ence is 77° 36', and the direction changing from N E to S E 
gives for the second line a bearing of S 77° 36' E. 

(652) To the bearing of the given line, viz. , N 13° 15' W, 
we add the angle 40° 20', which is turned to the left. The 
sum is 53° 35', which gives for the second line a bearing of 
N 53° 35' W. 

(653) The bearing of the first course, viz., S 10° 15' W, 
is found in the column headed Mag. Bearing, opposite Sta. 0. 
For the first course, the deduced or calculated bearing 
must be the same as the magnetic bearing. At Sta. 4 -j- 40, 



•248 



SURVEYING. 



an angle of 15° 10' is turned to the right. It is at once evi- 
dent that if a person is traveling in the direction S 10° 15' W, 
and changes his course to the right 15° 10', his course will 
approach a due westerly direction by the amount of the 
change, and the direction of his second course is found by 
adding to the first course, viz., S 10° 15', the amount of such 



Station. 


Deflection. 


Mag. Bearing. 


Ded. Bearing. 


54 + '25 






49 + 20 


L. 


35° 14' 


S 25° 40' W 


S 25° 39' W 


44 + 80 


L - 


10 = 47' 


S 50° 50' W 


S 50° 53' W 


33 + 77 


R. 


16 c 55' 


S 61°45'W 


S GV 40' W 


25 + 60 


R. 


24° 40' 


S 44° 50' W 


S 44 c 45' W 


16 + 20 


L. 


15° 35' 


S 20° 00' W 


S 20 = 05' W 


8 + 90 


R. 


10° 15' 


S 35° 50' W 


S 35° 40' W 


4+40 


R. 


15 c 10' 


S 25° 20' W 


S 25° 25' W 







S 10° 15' W 


S 10° 15' W 



change in direction. The sum is 25° 25', and the second 
course S 25° 25' W. The needle at this point reads S 25° 20' \Y. 
The difference between the magnetic bearing and the cal- 
culated bearing may be owing to local attraction, but as we 
cannot read the needle to within 10 minutes, we must gen- 
erally ascribe small discrepancies to that cause. This cal- 
culated bearing we write in its proper column opposite Sta. 
4 + 40, where the change in direction occurred. 

The next angle is 10° 15' to the right, which we add to the 
previous calculated bearing S 25° 25' W, giving S 35° 40' W 
for the calculated bearing of the third course, which extends 
from Sta. 8 + 90 to 16 + 20. In a similar manner, the stu- 
dent will calculate the remaining bearings, considering well 
how the changes in direction will affect his relations to the 
points of the compass. A plat of the notes to a scale of 
400 ft. to the inch is given in Fig. 65. 



SURVEYING. 



249 



***v 



* 



^ 



. ^ 1/ 



33 WK.16°55' 



{$> 



^ 



^ 



t6+2o\jf J5 ° 35 ' 



Assume the starting point A, 
Fig. 05, which is Sta. 0, and 
through it draw a straight line. 
The first angle, viz., 15° 10' to 
the right, is turned at vSta. 4 -J- 40, 
giving for the first course a length 
of 440 ft. Scale off this distance 
from A to a scale of 400 ft. to 
the inch, locating the point B y 
which is Sta. 4 -f- 40, and write on 
the line its bearing S 10° 15' W, 
as recorded in the notes. Pro- 
duce A B to C, making B C greater 
than the diameter of the protrac- 
tor, and from B lay off to the 
right of B C the angle 15° 10'. 
Join this point of angle measure- 
ment with B by a straight line, 
giving the direction of the second 
course. The end of this course is 
at Sta. 8 -f 90, and its length is 
the difference between 8 -f- 90 and 
4 + 40, which is 450 ft. This dis- 
tance we scale off from B, Sta. 
4 -\~ 40, locating the point D, and 
write on the line its bearing of S 
25° 20' W, as found in the notes. 
We next produce B D to E, mak- 
ing D E greater than the diameter 
of the protractor, and at D lay 
off the angle 10° 15' to the right 
of D E, giving the direction of the 
next course. In a similar man- 
ner, plat the remaining notes given 
in Example 653. The student in 
his drawing will show the pro- 
longation of only the first three 
lines, drawing such prolongations 



250 



SURVEYING. 



in dotted lines. In platting the remaining angles, he 
will produce the lines in pencil only, erasing them as 
soon as the forward angle is laid off. Write the proper 
station number in pencil at the end of each line as soon as 
platted, and the angle with its direction, R. or L. , before 
laying off the following angle. Write the bearing of each 
line distinctly, the letters reading in the same direction in 
which the line is being run. The magnetic meridian is 
platted as follows: The bearing of the course from Sta. 
33+77 to Sta. 44 + 80 is S 61° 45' W, i. e., the course is 
Gl° 45' to the left of a north and south line, which is the 
direction we wish to indicate on the map. Accordingly, we 
place a protractor with its center at Sta. 33 + 77 and its 
zero on the following course, and read off the angle 61° 45' 
to the right. Through this point of angle measurement and 
Sta. 33 + 77 draw a straight line N S. This line is the 
required meridian. 

(654) Angle B — 39° 25'. From the principles of trigo- 
nometry (see Art. 1 243), we have the following proportion: 

sin 39° 25' : sin 60' 15' :: 415 ft. : side A B. 

sin 60° 15' = .8682. 
415 ft. X .8682 = 360.303 ft. 
sin 39° 25' = .63496. 
360.303 ^ .63496 = 567.442 ft., the side A B. Ans. 




C 

Fig. 66. 
and the line A B produced as required. 



(655) (See Fig. 66.) At 
A we turn an angle B A C of 
60° and set a plug at C 100' 
from A. At (7 we turn an 
angle A C B of 60° and set a 
plug at B 100' from C. The 
point B will be in the line 
A B, and setting up the in- 
strument at B, we turn the 
angle C B A = 60°. The 
instrument is then reversed, 



SURVEYING. 251 

(656) See Art. 1245. 

• (657) See Art. 1246. 

(658) See Art. 1246. 

(659) See Art. 1248. 

(660) See Art. 1249. 

(681) A 5° curve is one in which a central angle of 
5° will subtend a chord of 100 ft. at its circumference. Its 
radius is practically one-fifth of the radius of a 1° curve, 
and equal to 5,730 ft. -^5 = 1,146 ft. 

(662) The degree of curve is always twice as great as 
the deflection angle for a chord of 100 feet. 

(663) See Art. 1249 and Fig. 282. 

(664) Formula 90, C= 2 R sin D. (See Art. 1250.) 

(665) Formula 91, T— R tan \ I. (See Art. 1251.) 

(666) The intersection angle C E F, being external to 
the triangle A E C, is equal to the sum of the opposite 
interior angles A and C. A .= 22° 10' and C = 23° 15'. 
Their sum is 45° 25' = C E F. Ans. 

The angle A E C— 180° - (22° 10' + 23° 15') = 134° 35'. 
From the principles of trigonometry (see Art. 1243), 
we have 

sin 134° 35' : sin 23° 15' :: 253.4 ft. : side A E; 
whence, side A E — 140.44 ft., nearly. Ans. 
Also, sin 134° 35' : sin 22° 10' :: 253.4 ft. : side CE; 
whence, side C E — 134.24 ft., nearly. Ans. 

(667) We find the tangent distance T by applying 
formula 91, 7^=7? tan J L (See Art. 1251.) From the 
table of Radii and Deflections we find the radius of a 6° 15' 

curve = 917.19 ft.; \ / = 3& 10 = iy ° 35 > ; tan 17 ° 35 > _ 

z 

.3169. Substituting these values in the formula, we have 
T= 917.19 X .3169 = 290.66 ft. Ans. 

(668) We find the tangent distance T by applying 
formula 91, T=R tan % I. (See Art. 1251.) From the 



B52 SURVEYING. 

table ot Radii and Deflections we find the radius of a 3° 15 

curve is 1,763.18 ft.; | /= U 12 = 7° 06'; tan 7° 06' = 

.12450. Substituting these values in the above formula, we 
have T- 1,763 X .12156 = 219.62 ft. Ans. 

(669) See Art. 1252. 

(670) The angle of intersection 30° 45', reduced to 
decimal form, is 30.75°. The degree of curve 5° 15', reduced 
to decimal form, is 5.25°. Dividing the intersection angle 
30.75° by the degree of curve 5.25 (see Art. 1252), the 
quotient is the required length of the curve in stations of 

100 ft. each. y ' Q = 5.8571 full stations equal to 585.71 ft. 

5.20 

(671) In order to determine the P. C. of the curve, we 
must know the tangent distance which, subtracted from the 
number of the station of the intersection point, will give us 
the P. C. We find the tangent distance T by applying 
formula 91, T—R tan£7. (See Art. 1251.) From the 
table of Radii and Deflections we find the radius of a 5° curve 

is 1,146.28 ft.; \I= 33 ° 6 = 16° 33'; tan 16° 33' = .29716. 

P.C.5°R. Substituting these val- 

A \ g Tz 340.63 ,Sta '20+37.8 ues in formula 91, we 

i6 + d^e^^f^;55 (>6' have T = 1,146.28 X 

%\Z) .29716 = 340.63 ft. In 

23+59JZ^' T ' Fig. 67, let A B and 

/ C C D be the tangents 

' j which intersect in the 

/ point E, forming an 

fig. er. angle D E F = 33° 06'. 

The line of survey is being run in the direction A B, and 

the line is measured in regular order up to the intersection 

point E, the station of which is 20 -f- 37.8. Subtracting the 

tangent distance, BE— 340.63 ft. from Sta. 20+ 37.8, we 

have 16 + 97.17, the station of the P. C. at B. The inter 

section angle 33° 06' in decimal form is 33.1°. Dividing 



SURVEYING. 253 

this angle by 5, the degree of the curve, we obtain the 

33. 1 

length B G D of the curve in full stations. — = —= 0.62 

stations = G62 ft. The length of the curve, GG2 ft., added 
to the station of the P. C. , viz. ,16+97.17, gives 23 + 59. 17, 
the station of the P. T. at D. 

(672) The given tangent distance, viz., 291.16 ft., was 

obtained by applying formula 91, T=R tan \ I (see 

Art. 1251), 7=20° 10', and } /== 10° 05', tan 10° 05' = 

.17783. Substituting these values in the above formula, we 

291 16 
have 291. 16 = R X . 17783 ; whence, R = — ^ - = 1,637. 29 ft. 

.177oo . 

Ans. 

The degree of curve corresponding to the radius 1,637.29 

we determined by substituting the radius in formula 89, 

50 
R = - — p: (see Art. 1249), and we have 
sin D v ' 

K() K() 

1,637.29 = -A-^; whence, sin£> = r—^— = .03054. 
sin D 1,637.29 

The deflection angle corresponding to the sine .03054 is 
1° 45', and is one-half the degree of the curve. The degree 
of curve is, therefore, 1° 45' X 2 = 3° 30'. Ans. 

(673) Formula 92, d= ~ (See Art. 1255 and 
Fig. 283.) 

(674) The ratio is 2; i. e., the chord deflection is double 
the tangent deflection. (See Art. 1254 and Fig. 283.) 

(675) As the degree of the curve is 7°, the deflection 

angle is 3° 30' = 210' for a chord of 100 ft., and for a chord 

210' 
ot 1 ft. the deflection angle is — — = 2. 1' ; and for a chord of 

48.2 ft. the deflection angle is 48.2x2. l' = 101.22' = l° 41.22'. 

(676) The deflection angle for 100-ft. chord is — ^— = 
3° 07V = 187.5', and the deflection angle for a 1 -ft. chord is 



254 SURVEYING. 

i^- 5 - = 1.875'. The deflection angle for a chord of 72.7 ft. 
is, therefore, 1.875' X 72.7 = 136.31' = 2° 16.31'. 

(677) We find the tangent deflection by applying 

formula 93, tan def. = — ^. (See Art. 1255.) c — 50. 

50 2 = 2,500. The radius R of 5° 30' curve = 1,042.14 ft. 
(See table of Radii and Deflections.) Substituting these 

values in formula 93, we have tan def. = ' = 1.199 

2,084. 28 

ft. Ans. 

(678) The formula for chord deflections is d= -^. (See 

Art. 1255, formula 92.) <r=35.2. 35. 2 2 = 1,239.04. The 

radius R of a 4° 15' curve is 1,348.45 ft. Substituting these 

1 239 04 
values in formula 92, we have d— t^t^t-l = .919 ft. Ans. 

1^ o4o. 45 

(679) The formula for finding the radius R is R = 

Kf) 

(See Art. 1 249.) The degree of curve is 3° 10'. D, 



sin U 



the deflection angle, is ——- = 1° 35'; sin 1° 35' = .02763. 

Substituting the value of sin D in the formula, we have R = 

50 

; whence, R = 1,809.63 ft. Ans. 

The answer given with the question, viz., 1,809.57 ft., 
agrees with the radius given in the table of Radii and 
Deflections, which was probably calculated with sine given to 
eight places instead of five places, as in the above calculation, 
which accounts for the discrepancy in results. 

(680) In Fig. 68, let A B and A C represent the given 
lines, and BC the amount of their divergence, viz., 18.22 

B ft. The lines will form a 

A^^^zz^T^J^Z Z-D*£' 22 * triangle ABC, of which 

4y the angle A — 1°. Draw 

fig. C8. a perpendicular from A to 

Z>, the middle point of the base. The perpendicular will 



SURVEYING. 255 

bisect the angle A and form two right angles at the base of 
the triangle. In the triangle A D B we have, from rule 5, 

Art. 754,tan^£> = ^v B A D = 30', B D = l! ' 



/4Z7 ' 2 

9.11 ft., and tan 30' = .00873. Substituting known values 

9 11 
in the equation, we have .00873 = -jrjy, whence, AD — 

5^| = 1,043.53 ft. An, 

By a practical method, we determine the length of the 
lines by the following proportion : 

1.745 : 18.22 :: 100 ft. : the required length of line; 
-j 090 
whence, length of line = — ^r— - = 1,044.13 ft. Ans. 
& 1.745 

The second result is an application of the principle of two 
lines 100 ft. in length forming an angle of 1° with each 
other, which will at their extremity diverge 1.745 ft. 

, flC1 , ~ - 24° 15' 24.25° , . 

(681) Degree of curve = -g-— = ^^ = 4 . Ans. 

(682) See Arts. 1264, 1265, 1266, and 1267. 

(683) See Arts. 1 269 -1 274. 

(684) Denote the radius of the bubble tube by x\ the 
distance of the rod from the instrument, viz., 300', by d; 
the difference of rod readings, .03 ft., by Ji, and the move- 
ment of the bubble, viz., .01 ft., by 5. By reference to Art. 
1275 and Fig. 289, we will find that the above values have 
the proportion h:S. \\d\x. Substituting known values 

in the proportion, we have .03 : .01 :: 300 : x; whence, 

3 
x = — = 100 ft., the required radius. Ans. 

. Uo 

(685) See Art. 1277. 

(686) See Art. 1278. 

(687) To the elevation 61.84 ft. of the given point, we 
add 11.81 ft., the backsight. Their sum, 73.65 ft., is the 
height of instrument. From this H. I., we subtract the fore- 



256 SURVEYING. 

sight to the T. P., viz., 0.49 ft., leaving a difference of 73.16 
ft., which is the elevation of the T. P. (See Art. 1279.) 

(688) See Art. 1280. 

(689) See Art. 1281. 

(690) See Art. 1282. 

(691) See Art. 1286. 

(692) See Art. 1289. 

(693) See Art. 1290. 

(694) See Art. 1291. 

(695) The distance between Sta. 66 and Sta. 93 is 27 
stations. As the rate of grade is -f 1-25 ft. per station, the 
total rise in the given distance is 1.25ft. X 27 = 33.75 ft., 
which we add to 126.5 ft., the grade at Sta. 66, giving 160.25 
ft. for the grade at Sta. 93. (See Art. 1291.) 

(696) See Art. 1292. 

/ ftQ7 v -16.4' -10. 3' 4-11.4' 

(697) —^r- -^r- 



Contour 50.0 at 46.0 ft. to left of Center Line. £ 
Contour 40.0 at 94.0 ft. to left of Center Line. £ 
Contour 30.0 at 128.0 ft. to left of Center Line. 



84' 96' 

Contour 60.0 at 26 ft. to right of Center Line. 
Contour 70.0 at 106.9 ft. to right of Center Line. 
Elevation 76.7 at 180 ft. to right of Center Line, 



(698) The elevations of the accompanying level notes 
are worked out as follows: The first elevation recorded in 
the column of elevations is that of the bench mark, abbre- 
viated to B. M. This elevation is 161.42 ft. The first rod 
reading, 5.53 ft., is the backsight on this B. M., a plus read- 
ing, and recorded in column of rod readings. This rod 
reading we add to the elevation of the bench mark, to deter- 
mine the height of instrument, as follows: 161.42 ft. -J- 5.53 
ft. = 166.95 ft., the H. I. The next rod reading, which is 
at Sta. 40, is 6.4 ft. The rod reading means that the sur- 
face of the ground at Sta. 40 is 6.4 ft. below the horizontal 
axis of the telescope. The elevation of that surface is, 
therefore, the difference between 166.95 ft., the H. I., and 6.4 

ft, the rod reading. 166.95 —6.4 = 160.55 ft. The— — ft. isa 



SURVEYING. 



257 



fraction so small that in surface elevations it is the univer- 
sal "practice to ignore it, and the elevation of the ground at 



Station. 


Rod 

Reading. 


Height 
Instrument. 


Elevation. 


Grade. 


B. M. 


+ 5.53 


166.95 


161.42 




40 


6.4 




160.5 


162.0 


41 


7.2 




159.7 


160.485 


41 + GO 


10.9 




156.0 




42 


8.6 




158.3 


158.97 


43 


8.8 




158.1 


157.455 


T. P. - 


8.66 




158.29 




+ 


2.22 


160.51 






44 


4.8 




155.7 


155.94 


45 


6.3 




154.2 


154.425 


46 


8.8 




151.7 


152.91 


47 


9.9 




150.6 


151.395 


48 


11.1 




149.4 


149.88 


T. P. - 


11.24 




149.27 




+ 


3.30 


152.57 






49 


4.7 




147.9 


148.365 


50 


7.1 




145.5 


146.85 


51 


8.7 




143.9 


145.335 


52 


9.8 




142.8 


143.82 


53 


10.9 




141.7 


142,305 


T. P. - 


11.62 




140.95 





Sta. 40 is taken at 160.5 ft. The rod reading at St a. 41 is 
7.2, which, subtracted from 166.95 ft., gives for that station 
an elevation of 159.7. The remaining rod readings up to 



258 SURVEYING. 

and including that at Sta. 43, we subtract from the same 
H. I., viz., 166.95. Here at a T. P., a rod reading of 8.66 
ft. is taken and recorded in the column of rod readings. 
This reading being a foresight is minus, and is subtracted 
from the preceding H. I. This gives us for the elevation of 
the T. P., 166.95 ft. — 8 66 ft =158.29 ft., which we record in 
the column of elevations. The instrument is then moved for- 
wards and a backsight of 2.22 ft taken on the same T. P. and 
recorded in the column. This is a plus reading, and is added 
to the elevation of the T. P., giving us for the next H. I. 
an elevation of 158.29 ft. + 2.22 ft. = 160.51 ft. The next 
rod reading, viz., 4.8, is at Sta. 44, and the elevation at that 
station is the difference between the preceding H. I., 160.51, 
and that rod reading, giving an elevation of 160.51 ft. —4.8 
ft. = 155.7 ft., which is recorded in the column of elevations 
opposite Sta. 44. In a similar manner, the remaining ele- 
vations are determined. 

In checking level notes, only the turning points rod read- 
ings are considered. It will be evident that starting from 
a given bench mark, all the backsight or plus readings will 
add to that elevation, and all the foresight or minus read- 
ings will subtract from that elevation. If now we place in 
one column the height of the B. M., together with all the 
backsight or -|- readings, and in another column all the fore- 
sight or — readings, and find the sum of each column, then, 
by subtracting the sum of the — readings from the sum of 
the + readings, we shall find the elevation of the last point 
calculated, whether it be a turning point or a height of in- 
strument. Applying this method to the foregoing notes 
we have the following : 

4- readings. — readings. 

B. M. 16 1.42 ft. 8.66 ft. 

5.5 3 ft. 11.2 4 ft. 

2.2 2 ft. 11.6 2 ft. 

330 ft - 3L52 ft. 

1 7 2.4 ? ft. 
31.52 ft. 



140.9 5 ft. 



SURVEYING. 



259 



The difference of the columns, viz., 140.95 ft., agrees 
with the elevation of the T. P. following Sta. 53, which is 
the last one determined. A check mark \f is placed opposite 
the elevation checked, to show that the figures have been 
verified. The rate of grade is determined as follows : In one 
mile there are 5,280 ft. = 52. 8 stations. A descending grade 



of 80 ft. per mile gives per station a descent of 



80ft. 
52.8 



= 1.515 



ft. The elevation of the grade at Sta. 40 is fixed at 162.0 
ft. As the grade descends from Sta. 40 at the rate of 1.515 
ft. per station, the grade at Sta. 41 is found by subtracting 
1.515 ft. from 162.0 ft., which gives 160.485 ft., and the 
grade for each succeeding station is found by subtracting 
the rate of grade from the grade of the immediately 
preceding station. 

A section of profile paper is given in Fig, 69 in which the 
level notes are platted, and upon which the given grade line 













i 






























1 
























































































































\r 


^. ^ 


^ 


Iri 


























\/ 






e <e<°.~, 


































~~y~i 




















£> 












*..-*• c r* 
































•^zP 


















—ft 
















&/0*r<- 






mev-. 


150. 


f 




so 




























*jH 


















^^^ 














ai 




















"^vT" 












— -e 






















v ~~- 


-■ »^^ 




»o 




— ? 
























~~^ — ■ 


Z^~-~~~. 


o 
































OJ 




^b 




























































t-Nfc 
































^* 


































*i 




























*n 
































il 
































































^ 
































>* 
































































U 
































v. 
































tft 






























































































































5 

































































































































40 Fig. 69. SO 

is drawn. The profile is made to the following scales: viz., 
horizontal, 400 ft. = 1 in. ; vertical, 20 ft. = 1 in. 

Every fifth horizontal line is heavier than the rest, and 
each twenty-fifth horizontal line is of double weight. Every 
tenth vertical line is of double weight. The spaces between 
the vertical lines represent 100 ft., and those between the 



G. G. IV.— J 6 



260 SURVEYING. 

horizontal lines 1 ft. The figure represents 1,500 ft. in length 
and 45 ft. in height. Assume the elevation of the sixth 
heavy line from the bottom at 150 ft. The second vertical 
line from the left of the figure is Sta. 40, which is written 
in the margin at the bottom of the page. Under the next 
heavy vertical line, ten spaces to the right, Sta. 50 is writ- 
ten. The elevation of Sta. 40, as recorded in the notes, is 
160.5 ft. We determine the corresponding elevation in 
the profile as follows: 

As the elevation of the sixth heavy line from the bottom 
is assumed at 150 ft., 160.5 ft., which is 10.5 ft. higher, must 
be 10|- spaces above this line. This additional space covers 
two heavy lines and one-half the next space. This point is 
marked in pencil. The elevation of Sta. 41, viz., 159.7 ft., 
we locate on the next vertical line and 9.7 spaces above the 
150 ft. line. The next elevation, 156.0 ft., is at Sta.' 41 + 
60. This distance of 60 ft. from Sta. 41 we estimate by the 
eye and plat the elevation in its proper place. In a similar 
manner, we plat the remaining elevations and connect the 
points of elevation by a continuous line drawn free-hand. 
The grade at Sta. 40 is 162.0 ft. This elevation should be 
marked in the profile by a point enclosed by a small circle. 
At each station between Sta. 40 and Sta. 53 there has been 
a descent of 1.515 ft., making a total descent between these 
stations of 1.515 X 13 = 19.695. The grade at Sta. 53 will, 
therefore, be 162.0 ft. - 19.695 ft. = 142.305 ft. Plat the 
elevation in the profile at Sta. 53, and enclose the point in a 
small circle. Join the grade point at Sta. 40 with that at 
Sta. 53 by a straight line, which will be the grade line 
required. Upon this line mark the grade — 1.515 per 
100 ft. 

+ 11° 



— 5° — 9° 

(699) -fr W 



120' 



Nine 5-foot contours are included within the given slopes, 
as follows: 



Contour 70.0 at 32.0 ft. to left of Center Line. ^ 
Contour 65.0 at 64.0 ft. to left of Center Line. -S 
Contour 60.0 at 96.0 ft. to left of Center Line. »- 
Contour 55.0 at 136.0 ft. to left of Center Line, c 
Elevation 51.0 at 182.0 ft. to left of Center Line. U 



Contour 80.0 at 26.0 ft. to right of Center Line. 
Contour 85.0 at 52.0 ft. to right of Center Line. 
Contour 90.0 at 78.0 ft. to right of Center Line. 
Contour 95.0 at 104.0 ft. to right of Center Line. 
Elevation 98.1 at 120.0 ft. to right of Center Line. 



SURVEYING. *61 

(700) 1.745 ft. X 3 = 5.235 ft., the vertical rise of a 3° 

slope in 100 ft., or 1 station and - — - =1.91 stations = 191 ft. 

5.235 

(701) (See Question 701, Fig. 15.) From the instru- 
ment to the center of the spire is 100 ft. -f- 15 = 115 ft., and 
we have a right triangle whose base A D = 115 ft. and angle 
A is 15° 20'. From rule 5, Art. 754, we have tan 45° 20' = 

5l%^ ; whence, 1.01170 = =^; or B D = 116.345 
115 115 

feet. The instrument is 5 feet above the level of the base; 

hence, 116.345 ft. -j- 5 ft. = 121.345, the height of the spire. 

(702) Apply formula 96. 

Z = (log h - log H) X 60,384.3 X (l + - + ^~ 64 ° ), 

(See Art. 1304.) 

log of k, 29.40 = 1.46835 
log of//, 26.95 = 1.43056 

Difference = 0.03779 

t+t' - 64° . , 74 + 58- 64 



= 1 + '—r— = 1.0755. 



900 ' 900 

Hence, Z= .03779 X 60,384.3 X 1.0755 = 2,454 ft., the 
difference in elevation between the stations. 

(703) See Art. 1305. 

(704) See Art. 1308. 

(705) See Art. 1308. 



SURVEYING AND MAPPING 



(QUESTIONS 706-734.) 



(706) See Art. 1310 and Fig. 306. 

(707) See Art. 1311 and Figs. 307 and 308. 

(708) As the bearing of the line A B is N E, the end B 
will be east of the meridian passing through A. The depar- 
ture of A B is the distance which B is east of A, or of the 
meridian passing through A. Now, if from B we drop a 
perpendicular B C upon that meridian, B C 

will be the departure of A B. The latitude 
of A B is the distance which the end B is 
north of the end A. The distance A C, meas- 
ured on the meridian from A to the foot of 
the perpendicular from B, is the latitude of 
A B. 

From an inspection of Fig. 70, we see that 
the line A B, together with its latitude A C 
and departure B C, form a right triangle, 
right angled at C, of which triangle B C 
is the sin and A C the cos of the bearing 
30°. From rule 3, Art. 754, we have 
A C 




Fig. 70. 



cos A = -j—^ ; whence, A C = A B cos A ; and from rule 1, 
BC 



AB 
Art. 754, sin A 



whence, B C = A B sin A, and we 



AB' 

deduce the following: 

Latitude == distance X cos bearing. 
Departure = distance X sin bearing. 

(709) See Art. 1313. 

For notice of copyright, see page immediately following the title pag< 



264 SURVEYING AND MAPPING. 



(710) 








Bearing. 


Distances. 


Latitudes. 


Departures. 


23i° 


400 ft. 


367 5 


157 9 




20 ft. 


18 38 


07 89 




3 ft. 
423 ft. 


2 7 5 6 


1184 




3 8 8.636 ft. 


16 6.9 7 4 ft. 



We divide the distance 423 ft. into three parts, viz., 400 
ft., 20 ft., and 3 ft. If now we find the latitude and 
departure for 4 ft. and multiply them by 100, we shall obtain 
the latitude and departure for 400 ft. The latitude of 4 ft. 
is 3.675 ft., and the departure 1.579 ft. We place these 
figures under their proper headings as whole numbers. The 
latitude and departure of 2 ft. are 1.838 and 0.789, respect- 
ively, which we place as whole numbers under their proper 
headings, but removed one place to the right of the 
figures above them, as they are the latitude and departure 
of tens of feet. The latitude and departure of 3 ft. are 
2.756 ft. and 1.184 ft., respectively, and we place them under 
their proper headings, but removed one place to the right, 
as they are for units of feet. We now add up the partial 
latitudes and departures, and from the right of each sum 
we point off three decimal places, the same number as 
given in the traverse table, giving us for the required 
latitude 388.636 ft., and for the required departure 
166.974 ft. 



(711) 








Bearing. 


Distances. 


Latitudes. 


Departures. 


40° 


200 ft. 


1532 


12 86 




20 ft. 


15 32 


128 6 




5 ft. 
225 ft. 


3 8 30 


3214 




1 7 2,3 5 ft. 


1 4 4.6 7 4 ft. 



For the given bearing of 40° and distance of 225 ft., the 
latitude is 172.35 ft., and the departure 141.674 ft. 

The complement of the given bearing is the difference 



SURVEYING AND MAPPING. 205 

between 90° and 40°, which is 50°. With this complement 
as the bearing, we have 



earing. 


Distances. 


Latitudes. 


Departures. 


50° 


200 ft. 


12 8 6 


1532 




20 ft. 


128 6 


1532 




5 ft. 
225 ft. 


3214 


3 8 30 




14 4.6 7 4 ft. 


1 7 2.3 5 ft. 



in which the latitude and departure are exactly the reverse 
of those when the line had a bearing of 40°, the comple- 
ment of 50°. 

(712) See Art. 1314. 

(713) We rule 11 columns, headed as below. The lati- 
tudes and departures for the several courses we calculate by 
traverse tables; placing the north latitudes, which are -|-, in 
the column headed N -j-, and the south latitudes, which are 
— , in the column headed S— ; the cast departures, which 
are -j-, in the column headed E -{-, and the west departures, 
which are — , in the column headed W — . These several 
columns we add, placing their sums at the foot of the 
columns. The sum of the distances is 37.20 chains; the 
sum of the north latitudes 13.19 chains, and of the south 
latitudes 13. 16 chains. The difference is . 03 chain, or 3 links. 
The sum of the east departures is 12.60 chains, and the sum 
of the west departures is 12.56 chains. The difference is 
.04 chain, or 4 links. 

This difference indicates an error in either the bearings 
or measurements of the line or both. For had the work 
been correct, the sums of the north and south latitudes 
would have been equal. (See Art. 1314.) The correc- 
tions for latitudes and departures are made as shown in the 
following proportions, the object of such corrections being 
to make the sums of the north and south latitudes and of 
the east and west longitudes equal, and is called balancing 
the survey. (See Art. 1315.) 



266 



SURVEYING AND MAPPING. 



Sta- 
tions. 


Bearings. 


Distances. 


Latitudes. 


De- 
partures. 


Corrected 
Latitudes. 


Corrected 

De- 
partures. 




N + 


S — 


E + 


w — 


N + 


S — 


E + 


w— 


1 
2 
3 
4 


N 31i° W 
N 62° E 
S 36° E 

S 45F W 


10.40 ch. 

9.20 ch. 

7.60ch. 
10.00 ch. 


8.87 
4.32 


6.15 
7.01 


8.13 
4.47 


5.43 
7.13 


8.86 
4.31 


6.15 
7.02 


8.12 
4.46 


5.44 
7.14 






37.20 


13.19 


13.16 


12.60 


12.56 


13.17 


13.17 


12.58 


12.58 



Difference between N and S latitudes = .03 chain = 3 
links. 

Difference between E and W departures = .04 chain = 4 



links. 






Corrections for Latitudes. 


Corrections for Departures. 


37.20 : 10.40 :: 3 : 1 link. 


37.20 


10.40 :: 4 : 1 link. 


37.20 : 9.20 :: 3 : 1 link. 


37.20 


9.20 :: 4 : 1 link. 


37.20 : 7.60 :: 3 : link. 


37.20 


7.60 :: 4 : 1 link. 


37.20 : 10.00 :: 3 : 1 link. 


37.20 


10.00 :: 4 : 1 link. 



Taking the first proportion, we have 37.20 ch., the sum of 
all the distances : 10.40 ch., the first distance :: 3 links, the 
total error : 1 link, the correction for the first distance. 
The latitude of the first course, viz., 8.87 ch., is north, and 
as the sum of the north latitudes is the greater, we subtract 
the correction, leaving 8.86 chains. The correction for the 
latitudes of the second course is 1 link, and is likewise sub- 
tracted. The correction for the third course is less than 1 
link, and is ignored. The correction for the latitude of the 
fourth course is 1 link, and as the sum of the south latitudes 
is less than the north latitudes, we add the correction. We 
place the corrected latitudes in the eighth and ninth col- 
umns. In a similar manner we correct the departures, as 
shown in the above proportions, placing the corrected 
departures in the tenth and eleventh columns. 

(714) We rule three columns as shown below, the first 
column for stations, the second for total latitudes from Sta. 



SURVEYING AND MAPPING. 



267 



2, and the third for total departures from Sta. 2. Station 2 
being a point only, its latitude and departure are 0. The 
latitude of the second course, i.e., from Sta. 2 to Sta. 3, 
is 4- 4.31 chains, and the departure -f 8.12 chains. These 
distances we place opposite Sta. 3, in their proper columns. 
The latitude of the third course, i.e., from Sta. 3 to Sta. 4, 
is — 6.15 chains, and the departure -j- 4.46 chains. There- 
fore, the total latitude from Sta. 2 is the sum of -|- 4. 31 
and — 6.15, which is — 1.84 chains. The total departure 
from Sta. 2 is the sum of -j- 8. 12 and + 4. 46, which is -f 12. 58 
chains. These totals we place opposite Sta. Jf. in their 
proper columns. The latitude of the fourth course, i.e., 
from Sta. 4- to Sta. i, is — 7.02 chains, and the departure 

— 7.14 chains. These quantities we add with their proper 
signs to those previously obtained, which give us the total 
latitudes and departures from the initial Sta. 2, and we 
have, for the total latitude of Sta. 1, the sum of — 1.84 
and — 7.02, which is — 8.86 chains, and for the total depar- 
ture the sum of -J- 12.58 and — 7.14, which is -f- 5.44 chains. 
The latitude of the first course, i. e., from Sta. 1 to Sta. 2, 
is -f- 8.86 chains, and the departure — 5.44 chains. These 
quantities we add with their proper signs to those already 
obtained, giving us the total 
latitudes and departures from 
Sta. 2, and we have for the 
total latitude of Sta. 2 the sum 
of - 8.86 and + 8.86, which 
is 0; and for the total depar- 
ture, the sum of -j- 5.44 and 

— 5.44, which is 0. The lati- 
tude and departure of Sta. 2 
coming out equal to 0, proves 
the work to be correct. (See 
Art. 1316.) A plat of this 
survey made from total lati- 
tudes and departures from 

Sta. 2 is given in Fig. 71. Through sta. z draw ; men 
dian N S. Lay off on this meridian above Sta. 2 the total 




268 



SURVEYING AND MAPPING. 



latitude at Sta. 3, viz., -f 4.31 chains, a north latitude, and 
at its extremity erect a right perpendicular to the meridian, 
and upon this perpendicular scale off the total departure of 
Sta. 3, viz., -J- 8.12 chains, an east departure locating Sta. 3. 
A line joining Stations 2 and 3 will have the direction and 
length of the second course. For Sta. Jf. we have a total 
latitude of — 1.8-1 chains, a south latitude which we scale off 
on the meridian below Sta. 2. The total departure of this 
station is -J- -12.58 chains, which we lay off on a right per- 
pendicular to the meridian, locating Sta. Jf.. A line joining 
Stations 3 and 4 gives the direction and length of the third 
course. 



Stations. 



Total Latitudes 
from Station 2. 



Total 

Departures 

from Station 2. 



0.00 ch. 
+ 4.31 ch. 

- 1.84 ch. 

— 8.80 ch. 
0.00 ch. 



0.00 ch. 
+ 8.12 ch. 
+ 12.58 ch. 
+ 5.44 ch. 

0.00 ch. 



The total latitude of Sta. 1 is — 8.8G chains, a south lati- 
tude, which we scale off on the meridian below Sta. 2. The 
total departure of Sta. 1 is -|- 5.44 chains, an east departure, 
which we scale off on a right perpendicular to the meridian, 
locating Sta. 1. A line joining Stations 4. an d 1 will have 
the direction and length of the fourth course. The total 
latitude and departure of Sta. 2 being 0, a line joining Sta. 
1 with Sta. 2 will have the direction and length of the first 
course, and the resulting figure 2, 3, 4, 1 is the required 
plat of the survey. 

(715) See Art. 1318. 

(716) See Art. 1319. 

(717) vSee Art. 1320 and Fig. 313. 



SURVEYING AND MAPPING. 269 

(718) See Art. 1320. 

(719) The signs of the latitude always determine the 
character of the products, north or -|- latitudes giving north 
products and south or — latitudes giving south products. 
(See examples 722 and 723.) 

(720) See Art. 1322. 

(721) Rule twelve columns with headings as shown in 
the following diagram, calculate the latitudes and departures, 
placing them in proper order. Balance them, writing the 
corrected latitudes directly above the original latitudes, 
which are crossed out. Calculate the double longitudes from 
Sta. 2 by rule given in Art. 1319, and place them with 
their corresponding stations as shown in columns 11 and 12. 
Place the double longitudes in regular order in column 8. 
Multiply the double longitude of each course by the cor- 
rected latitude of that course, placing the products in col- 
umn 9 or 10, according as the products are north or south. 
Add the columns of double areas, subtracting the less from 
the greater and divide the remainder by 2. In this example 
the area is given in square chains, which we reduce to acres, 
roods, and poles as follows: Divide the sq. chains by 10, 
reducing to acres. Multiply the decimal 'part successively 
by 4 and 40, reducing to roods and poles. 



270 



SURVEYING AND MAPPING. 






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SURVEYING AND MAPPING. 273 

(724) See Art. 1326. 

(725) See Art. 1324. 

(726) Applying the trapezoidal rule (see Art. 1324), 
we have the number of spaces = - 6 -^ L = 32 ; since the end 
ordinates are zero, the mean ordinate is (4- -j- 7-J + 9f + 12 + 
124 + 16 + 19 -f 20f + 21i+ 25 + 25J- + 25f + 2G + 25| + 
24 -f 22 + 20i + 19 + 15 + 12* + Hi + 10 + 10 + 9£ + 9 + 
8f + SJ- + 7| + 6 + 2 J + 2) -f- 32 = 448.25 -r- 32, and the area 
is 640 X 448.25 -f- 32 = 8,965 square feet. Ans. 

(727) Applying formula 97, Art. 1 325, the volume is 

8,965 + 7,415 xl0Q = 
2 

819,000 cubic feet. Ans. 

(728) See Art. 1312. (a) The latitude of the line is 
equal to its length x cos of its bearing, or, letting L = 
latitude, L = 748 X cos 36° 42' = 748 X .80178 = 599.731 feet, 
which, since the course is northerly, is additive and is to be 
marked -\-. Ans. 

(b) Letting D denote the departure, we have D = 748 X 
sin 36° 42' = 748 X .59763 = 447.027 feet, which, since the 
course is easterly, is additive and is to be marked -f . Ans, 

(729) See note, Art. 1332. 

(730) See Art. 1334. 

(731) See Arts. 1341 to 1346. 

(732) See Art. 1346. 

(733) See Art. 1346. 

(734) See Arts. 1347 and 1351. 



STEAM AND STEAM ENGINES. 

(QUESTIONS 735-794.) 



(735) See Arts. 1367 to 1372. 

(736) See Arts. 1368 and 1369. 

(737) See Arts. 1378 and 1379. 

(738) See Arts. 1370 and 1371. 

(739) See Arts. 1375, 1374, and 1379. 

(740) (a) See Arts. 1380, 1370, and 1385. 

(b) To raise the temperature of the ice from 16° to 32° 
requires, according to formula lOl, Art. 1379, 

.504 X 1 X (32 - 16) = 8.064, say 8 B. T. U. 

To change the pound of ice at 32° to water requires 144 
B. T. U. (See Art. 1380.) To raise the temperature of 
the water from 32° to 212° requires 

1 X 1 X (212 - 32) = 180 B. T. U. 

To change the pound of water to steam requires, further, 
966 B. T. U. (See Art. 1381.) Hence, the total heat 
required is 8 + 144 + 180 + 966 = 1,298 B. T. U. 

(c) The sensible heat is 8 + 180 = 188 B. T. U. 

(d) The latent heat is 144 + 966 = 1,110 B. T. U. 

(741) (a) and (b). See Arts. 1376 and 1377. 

(c) IB. T. U. = 778 ft. -lb. 

30^ B. T. U. = 30£ X 778 = 23,729 ft. -lb. Ans. 

(742) 35 H. P. = 35 X 33,000 ft. -lb. per min. =35 X 

33,000 X 60 ft. -lb. per hour = 35 X 33 000 X 60 R T ^ per 

< 7o 
hour = 89,074.5 B. T. U. per hour. 

For notice of copyright, see page immediately following the title page. 

a. a. iv.— n 



276 STEAM AND STEAM ENGINES. 

But this is the heat actually used, or 20^ of the whole. 
Hence, the heat required is 89,074.5 '-=- .20 = 445,372.5 
B. T. U. per hour. Ans. 

(743) One horsepower = 33,000 X GO ft. -lb. per hour = 

33,000 X 60 _ „ TT , 

— B. T. U. per hour. 

7 i b 

Each pound of coal gives 14,000 B. T. U., of which 8$, or 
14,000 X .08 = 1,120 B. T. U. are utilized. Hence, the coal 
required per hour per H. P. is 

33 -°M* 60 -, 1,120 =2.27 lb. Ans. 

778 

(744) By formula IOI, Art. 1379, B-. T. U. = 

c W{t-t) =.2026 X 22i X (68-44) = 109.4 B. T. U. Ans. 

(745) {a) See Art. 1380. 

(/?) To raise the ice from 17° to 32° requires for each 
pound .504 X (32- 17) = 7.56 B. T. U. To melt it re- 
quires 144 B. T. U. Hence, 1 lb. requires 144 + 7.56 = 
151.56. 11 lb. requires 11 X 151.56 = 1,667.16 B.T.U. Ans. 

(746) By formula IOI, B.T.XJ.= c W(t- t) = .4805 X 
6 X (342 - 310) = 92.256 B. T. U. Ans. 

(747) Use formula 102, Art. 1384. The product of 
the weight, specific heat, and temperature of the copper is 
18 X .0951 X 305 = 522.099; of the iron, 13 X .1138 X 278 = 
411.2732; of the water, 32 X 1 X 56 = 1,792. The sum is 
2,725.3722. The product of the weight and specific heat of 
the copper is 18 X .0951 = 1.7118; of the iron, 13 X .1138 = 
1.4794; of the water, 32 X 1 = 32. The sum is 35.1912. 

2 725 3722 

Hence, the resulting temperature is ~\Z '\^~ = 77.45°. 

bo. 1912 

Ans. 

(748) (ci) 966.069 B. T. U. Ans. 

(b) To raise a pound from 63° to 212° requires 212 
— (13 = 149 B. T. U. To change it into steam requires 
966.069 more B. T. U. 966.069 + 149 = 1,115.069 B. T. U. 
for 1 lb. Hence, 8 X 1,115.069 = 8,920.552 B. T. U. are 
required. Ans. 



STEAM AND STEAM ENGINES. 277 

(749) To change 1 lb. of ice from 23° to 32° requires 
(32 — 23) X .504= 4.536 B. T. U. To melt the ice requires 
144 B. T. U. To change the water at 32° to steam at 212° 
requires 1,146.6 B. T. U. per pound. (See table of the 
Properties of Saturated Steam.) 1,146.6 -f 144 -f 4.536 = 
1,295.136 B.T. U. per pound. For 2.2 lb., 1,295.136 X 2.2 = 
2,849.3 B. T. U. are required. Ans. 

(750) {a) and {b) See Arts. 1387 and 1388. 

(c) Saturated steam at a given temperature has a definite 
pressure and can have no other ; that is, the pressure 
depends upon the temperature. A perfect gas at a given 
temperature may have any pressure; if the temperature re- 
mains constant, the pressure varies inversely as the volume. 

(751) See Arts. 1389 and 1391. 

(752) The volume of a pound at 80 lb. pressure, abso- 
lute, is 5.358 cu. ft. At 82 the volume is 5.235 cu. ft. 
Therefore (see table of the Properties of Saturated Steam), 

at 81 lb. pressure the volume is — — '-^— = 5.2965 cu. ft. 

5.2965 X 6 = 31.779 cu. ft. Ans. 

(753) From the table of the Properties of Saturated 
Steam, the absolute pressure is 60 lb. per sq. in., nearly. 
Hence, the gauge pressure is 60 — 14. 7 = 45.3 lb. per sq. 
in. Ans. 

(754) The steam gives up its latent heat and also cools 
from 310.123° to 135°. Hence, each pound gives up 
896.359 -f (310.123°- 135) = 1,071.482 B. T. U. But, to 
raise the temperature of the water requires 250 X (135 — 50) = 
21,250 B. T. U. 21,250 -4- 1,071.482 = 19.83 lb. Ans. 

(755) Referring to the table of the Properties of 
Saturated Steam, 1 lb. of steam at 350° occupies a volume 
of 3.272 cu. ft. Q$ X 3.272 = 20.041 cu. ft. = volume of 
6-J lb. of steam. Ans. 

(756) To evaporate the pound of water takes 966.069 
B. T. U 14,500 4- 966.069 = 15 lb. of water per lb. of coal. 

Ans. 



278 



STEAM AND STEAM ENGINES. 



(757) Referring to the table of the Properties of Satu- 
rated Steam, it is found that to raise 1 lb. of water from 32° 
and evaporate it at 841b. pressure, requires 1,178.091 B.T.U. 
22,000 -f- 1,178.091 = 18.675 lb. Ans. 

(758) (a) See Art. 1395. 

(b) It is not necessary to know either the diameter or 
length of cylinder, as they are both included in the volume. 

(759) (a) By formula 103, Art. 1395, 

W= 114/ V— 144 X 21 X 24 = 72,576 ft.-lb. Ans. 

(b) Work per minute = 72,576 X 60 ft.-lb. Hence, horse- 

72,576 X 60 



power 



33,000 



= 131.96 H. P. Ans. 



(760) (a) p = .57 X 60 = 34.2 lb. per sq. in. V=2.9 X 
1.3 = 3.77 cu. ft. By formula 103, work done per stroke = 
W= 144/ V= 144 X 34.2 X 3.77 = 18,566 ft.-lb. Ans. 

(b) Work per minute = 18,566.5 X 74 ft. -lb. Hence, horse 
18,566.5 X 74 _ 

P ° Wer= 33,000 = 41-*> H - P - Ans ' 

(761) See Art. 1404. 

(762) See Art. 1406. 

(763) See Arts. 1408 and 1409. 

(764) See Art. 1387. 

(765) See Arts. 1416 and 1417. 

(766) See Arts. 1414 and 1415. 

(767) See Art. 1418. 

(768) See Arts. 1418, 1420, and 1423. 

(769) See Arts. 1423 and 1434. 

(770) See Arts. 1432 and 1433. 

(771) vSee Arts. 1428 to 1430. 

(772) (a) The construction is shown in Fig. 72. The 
distance E F— 3J-" = travel of valve. Upon E Fas a diam- 
eter, describe the circle. Draw M N parallel to E F and 



STEAM AND STEAM ENGINES. 



279 



r y (= lead) above it. Lay off angle E O G — 30° = angle 
of advance. Drop the perpendicular GR upon M N. Then, 




Fig. 72. 

G R is the required lap. Upon being measured, it is found 
to be if". Ans. 

(b) With G R as radius and center G, describe a circle in- 
tersecting line O G in K Then, OK— \\" = port open- 
ing. Ans. 

(773) {a) In case the angle of advance is 60°, lay off 
E O G' (Fig. 72) = 60°, and from G' drop a perpendicular 
G' R! upon MN. Then, G' R! = lap = 1-^" = ff", nearly. 

Ans. 



The increase of lap is, therefore, -f-J — ^f = \% = f " 



(b) The port opening is, in this case, OK' 
(774) See Arts. 1436 to 1438. 



*', nearly. 
Ans. 



280 



STEAM AND STEAM ENGINES. 



(775) 20 : 28 :: eccentric throw : valve, travel, 
or, 20 : 28 :: 4 : valve travel. 
4 X 28 



Hence, valve travel = 



20 



= 5.6 in. Ans. 



(776) See Art. 1439. 




Fig. 73. 



(777) The cases from a to h are shown in Fig. 7&. 

(778) See Art. 1443. 

(779) See Art. 1445. 

(780) See Art. 1446. 

(781) See Art. 1448. 

(782) Cee Arts. 1453 to 1455. 

(783) (a) See Arts. 1456 to 1458. 

(b) According to formula 104, Art. 1457, 
S+.i $ + .07 .375 + . 07 _. 445 



Real cut-off - 

1 -f i 

Ratio of expansion 



1.07 
1 



real cut-off .410 



.07 
1 



1.0' 



= .416. 
Ans. 



= 2.4. Ans. 



STEAM AND STEAM ENGINES. 281 



(784) (a) Apparent cut-off = ¥ 9 ¥ = .225. Ans. 

,,, ^ , rr S -\- i .225 4- .03 .255 njmi/t A 

(*) Real cut-off =, ^ = ^ = — = . 2476. Ans. 

1 0° 
(r) Number of expansions = -^rv = 4.03. Ans. 

(785) (a) The volume Voi the cylinder is .7854 X 18 2 X 
40 = 10,178.8 cu. in. Using formula 103, Art. 1395, 

10 1 7S S 
144/ V- 144 x 34.6 X ' „' = 29, 348. 87 ft. -lb. per stroke. 

15728 Ans. 

,.. 29,348.87 X 140 < 

<*>" 33700CT -=^ 1H ' Pl AnS - 

(786) (a) The steam in the cylinder at cut-off is that 
included in 9 in. of the length of the cylinder and steam in 
clearance. 

The clearance is 3^ of the cylinder volume = 10,178.8 X 
.03 = 305.364 cu. in. 

The volume displaced by piston up to cut-off is .7854 X 
18 2 X 9 = 2,290.23 cu. in. 

Volume of steam admitted per stroke = 305.364 + 
2,290.23 = 2,595.594 cu. in. 

At 100 lb., absolute, 1 cu. ft. of steam weighs .230293 lb. 

2 595 594 
Hence, the steam per stroke is ' ' X .230293 = .3459 lb. 

1,728 Ans. 

(b) To find the steam per H. P. per hour, multiply {a) by 
number of strokes per hour and divide by H. P., or 

.3459 X 140 X 60 -v- 124.51 = 23| lb. steam per H. P. per 
hour. Ans. 

(787) From column 6, table of Properties of Saturated 
Steam, 

W — .106345 lb. per cu. ft. for 44 lb. pressure. 

= .1017941b. per cu. ft. for 42 lb. pressure. 

.106345 + .101794 -„,„„„ r , , 

= = .10407 lb. per cu. It. for 43 lb. 

2 

pressure. 

Then, 38 cu. ft. weigh 38 X .10407 = 3.95 + lb. Ans. 



282 STEAM AND STEAM ENGINES. 

(788) Work per hour = 75 X 2 X 2 X 20 X GO X .18 

ft. -lb. = 04,800 ft.-lb. = 6 tv 80Q = 83.29 B. T. U. Ans. 

<'7o 

(789) (a) See Art. 1391. 
(6) 84.5 +14.7 = 99.2 j 

22.37 -f 14.7 = 37.07 [■ Absolute pressures. Ans. 
35 +14.7 = 49.7 J 

(790) See Art. 1386. 

(791) See Art. 1401. 

(792) 10 H. P. = 33,000 X 10 X GO = 19,800,000 ft. -lb. 

19,800,000 _ ,' n ._ m TT _, ... 
per hour = — ^ ■ = 2o,4o0 B. T. U. But this is one- 

fifth the total number required, which is 25,450x5 = 127,250. 

Therefore, '~ = 19. 9G lb. of wood are required per hour. 
6 > 375 Ans. 

(793) See Art. 1442. 

(794) In cases (a) and (c) the eccentric is behind the 
crank 90° — angle of advance. Hence, the angle between 
eccentric and crank is 90° — 37° = 53°. Ans. 

In cases (b) and (d) the eccentric is ahead of the crank 
90° + angle of advance = 90° + 37° = 127°. Ans. 



STEAM AND STEAM ENGINES. 

(QUESTIONS 795-844.) 



(795) See Arts. 1464 and 1465. 

(796) See Art. 1466. 

(797) See Arts. 1468 to 1476. 

(798) See Arts. 1467 and 1476. 

(799) See Arts. 1482 to 1485. 

(800) See Art. 1495. 

(801) (a) See Art. 1497. 

(b) By formula 106, 

, IR 17 X 210 KAK . A 

piston speed = — = — — = 595 ft. Ans. 

o 

(802) See Art. 1499. 

(803) See Art. 1498. 

(804) Admission, cut-off, release, and compression are 
all too late. The back pressure is also excessive. Add lap 
to the valve, shift the eccentric ahead on the shaft, and 
make the exhaust port or exhaust pipe larger. 

(805) See Art. 1501. 

(806) See Arts. 1503 and 1509. 

(807) See Arts. 1504 and 1507." 

(808) See Arts. 1511 and 1514. 

(809) Absolute pressure of entering steam is 65 -f 
14.7 = 79.7 lb. per sq. in. ; of exhaust, 14.7 + 2.3 = 17 lb. 

Temperature of entering steam (from table of the Prop- 
erties of Saturated Steam) is 311.0°, nearly. 

For notice of copyright, see page immediately following the title page. 



284 



STEAM AND STEAM ENGINES. 



Temperature of exhaust steam is 210.45°. 
Absolute temperature T x of entering steam is 311.6 
460 = 771.6°. 

Absolute temperature T 2 of exhaust is 219.45 -f- 4(30 

679.45°. 

T - T 771.6- 679.45 92.15 



Thermal efficiency = 



T. 



771.6 



771.6 



11.94$. Ans. 

(810) See Arts. 1515 to 1519. 

(811) The M. E. P. is found as shown in Fig. 74. 




Fig. 74. 



The middle ordinates are drawn as explained in Art. 
1493. 

These ordinates are measured and multiplied by the scale 
of the spring, which reduces them from inches to lb. per 
sq. in. In the figure the sum of the 20 ordinates is 707.2 
pounds. The mean ordinate, or, what is the same thing, 



the M. E. P., is, therefore, '' 



35.36 lb. Draw the 



vacuum line at a distance of 



14.7 
40 



= .3675" below the atmos- 



pheric line M N. Choose the point A near the point of 
release, and draw A B parallel to M N. The height B C of 



STEAM AND STEAM ENGINES. 285 

this line above the vacuum line is f " ; therefore, the absolute 
pressure at A or B is f X 40 = 30 lb. The length A B = 
/= 2. 84", and the length of the diagram L is 3.6". The 
weight of a cu. ft. of steam at 30 pounds pressure, absolute, 
is .074201 lb. 

Substituting these values in formula 107, Art. 1507, 

13, 750 IW 13,750 X 2.84 X .074201 



Q 



PL 35.36 X 3.6 

22.76 lb. per I. H. P. per hour. Ans. 



(812) (a) See Art. 1477. 

(b) The three principal uses are the following : 

1. To find the I. H. P. of the engine. 

2. To detect defects in valve setting, and to serve as a 
guide in setting the valves. 

3. To determine, approximately, the steam consumption 
of the engine. 

(813) (a) Work per stroke per sq. in. of piston = 

M^X 40 = 69.4 ft. -lb. 

I/O 

Area of piston = 16 2 x .7854 = 201.0624 sq. in. 
Then, total work per stroke = 69.4 X 201.0624 = 13,953.73 
ft. -lb. Ans. 

(b) 120 rev. per min. = 120 X 2 = 240 strokes per min. 

Then, H. P. = 18 ' 96 a V£? U ° = 101-48H. P. An, 

oo, UUU 

(814) See Arts. 1522 and 1523. 

(815) (a) Using formula 105, Art. 1495, 

= PLAN = 47.1 X |f X ll a X .7854 X 210 X 2 
' ' ' 33,000 ' 33,000 

85.45 H. P. Ans. 

(b) 85.45 X .83 = 70.92 actual H. P. Ans. 

(c) 85.45 - 70.92 = 14.53 friction H. P. Ans, 



286 STEAM AND STEAM ENGINES. 

(816) (<?) GO pounds, gauge pressure, = GO -f- 14.7 = 
74.7 lb., absolute; 2 lb. above atmosphere = 2 -f- 14.7 = 
16.7 lb., absolute. 

Temperature of steam at 74.7 lb. pressure is, from the 
table of the Properties of Saturated Steam, 



306.^ 



X .3 



526 + ( 308.344- 306.526 ^ x7 = 3Qyi620 

Temperature of steam at 1G.7 is 

216.347 + (219.452 - 216.347) X .7 = 218.521°. Then, 

T x = 307.162° + 460° = 767.162°, 

and T a = 218. 521° + 460° = 678. 521°. 

™ . ~ . T t - 7\ 767.162 - 678.521 

Thermal efficiency = — 1 -= ' = ^ = 

7 , <o7.102 

.1155 = 11.55$. Ans. 

{b) In this case, the absolute pressure of the entering 
steam is 90 -f- 14.7 = 104.7 pounds per sq. in., and the pres- 
sure of the exhaust steam 3 pounds per sq. in., absolute. 

The temperature corresponding to the former pressure 

, ^ nn^nn /331. 169 - 327. 625\ 
is from the table = 331.169 — ( J 

330. 956°. The temperature corresponding to the latter pres- 
sure is 141.654°. Hence, T x is 330.956° + 460° = 790.956°, 
and 7; is 141. 654° + 460° = 601. 654°. 

Thermal efficiency = 79 °- 9 f " ^ L654 = .2393 = 23.93$. 

790. 9o6 Ans< 

(817) Area of high-pressure cylinder = 19 2 X .7854 sq. 
in. 

Area of low-pressure cylinder = 32 2 X .7854 = 804.25 sq. 
in. 

Ratio of area of high-pressure cylinder to area of low- 

,. . 19 2 X .7854 19 2 361 - 

pressure cylinder = ^ x ^ = —, = j^g- 

M, E. P. of high-pressure cylinder reduced to low-pressure 

OP 1 

cylinder = 52 X . = 18.332 lb. per sq. in. 



STEAM AND STEAM ENGINES. 287 

Total M. E. P. reduced to low-pressure cylinder = 
18 + 18.332 = 3(3.332 lb. per sq. in. 

(a) Substituting now, in formula 105, Art. 1495, 

T TT _ PLAN 36. 332 xffX 804. 25x120x2 KQ1 on 
L H - R = ^000- = ' 3370-00 = 6S1 '™' 

Ans. 

(b) Since the stroke is the same for each cylinder, the 
ratio of the work done by the two cylinders is proportional 
to the ratio of the two M. E. P's reduced to the low- 
pressure cylinder. Hence, the ratio of the work done in 
the high-pressure cylinder to that done in the low-pressure 
cylinder is 18.332 : 18 = 1.0184. Ans. 

(818) (a) See Art. 1501. 

(b) 1. Decrease the angular advance. 

2. Increase the angular advance. 

3. Lower the boiler pressure or decrease the number of 
revolutions. 

4. Raise the boiler pressure or increase the number of 
revolutions. 

(819) (a) The points desired are shown in Fig. 75. To 
locate the point of cut-off, prolong the steam and expansion 
lines till they intersect, as at 1. From 1, drop the perpen- 
dicular 1 a, and a will be the point of cut-off. 

The point of release may be located by prolonging the 
expansion curve, and noting the point where the actual 
curve departs from it, as shown at b. The point of com- 
pression c is easily located. 

(b) The M. E. P's of the two diagrams are found as 
shown in the figure. The length of each diagram is divided 
into 20 equal parts, and ordinates are erected at the middle 
points of the divisions. The lengths of these ordinates mul- 
tiplied by the scale of the spring used, in this case 1" = 30 
lb., added together and divided by 20 gives the required 



288 



STEAM AND STEAM ENGINES. 



M. 



E. P 

1 



The M. E. P. of diagram A is found to be 30.75 




Fig. ?5. 

lb. per sq. in., and the M. E. P. of diagram B is found to be 
44.6 lb. per sq. in. Ans. 

(820) Diameter of drivers = 80" = ff = 6f ft. 
Circumference of drivers = Gf X 3.141(3 = 20.044 ft. 
60 miles per hour = 1 mile per minute. 
One mile contains 5,280 ft. Hence, the number of revo- 
lutions of the driver per minute is ' 

1 20.944 

Area of piston = .7854 X 10' = 283.53 sq. in. 
Now, using formula 105, Art. 1495, 



•IX 



5,280 



I. H. P. 



PL A N 
33,000 



52. 5 X- 2 X 283. 53X^^X2 



33,000 



= 454.86. 



STEAM AND STEAM ENGINES. 280 

Since there are two cylinders, the total horsepower = 
454,86 X 2 = 909.72 I. H. P. Ans. 

(821) See Art. 1508. 

(822) The total heat of steam at 4|- pounds pressure is, 
from the table of the Properties of Saturated Steam, 

1 ' m " 641 1 1 - 131 " 46 ^ 1,130.051 B.T.U. 

2 

Now, using formula 108, Art. 1520, 

w= H-t, + Z% = 1,130.051 - 130 + 32 = ^ Q ^ ^ 

t x — r 2 130 — 55 

(823) (a) and (b) See Arts. 1472 and 1474. 

(824) (a) Efficiency = 12 . 325 = . 8024 = 80. 24^. Ans. 

15. do 

(b) Area of piston = .7854 X 9 2 = 63.617 sq. in. Stroke = 
12" = 1 foot. 

Using formula 105, Art. 1495, 

T ^ ^ PLAN 

I. H. P. X 33,000 _ 15.36 X 33,000 _ 1fifilh 

LAN ~~1X 63.617 X 240 xT b A 

Ans. 

(825) To obtain an idea of the condensation of the 
steam in the cylinder. There is generally more condensa- 
tion at cut-off than at release, on account of reevaporation, 
and if the steam consumption be calculated from both points, 
the difference will tell, more or less approximately, the 
amount of condensation at cut-off. 

(826) The scale of the spring should be, in general, 
not less than \ of the boiler pressure. 

(a) -^ 4 - = 27; hence, a 30 spring should be used. Ans. 

(b) J ^- 5 - = 57^-; hence, a 60 spring should be used. Ans. 

(c) 1.834" X 40 = 73.36 lb. per sq. in. Ans. 



200 STEAM AND STEAM ENGINES. 

(827) Total heat of steam at an absolute pressure of 
7 pounds is, from the table of the Properties of Saturated 
Steam, 1,135.908. Using formula 108, Art. 1520, 

1,135.908-118.06 + 32 - 

-? — — ±- — - x 906 = 13,580 lb. Ans. 

(828) The areas of the three pistons are proportional 

to the squares of their respective diameters. Hence, the 

M. E. P. of the high-pressure cylinder reduced to the low- 
o'ya 

pressure cylinder is 72 Xtt,= 12.05 lb. per sq. in. The 

M. E. P. of the intermediate cylinder reduced to the low- 

12 2 
pressure cylinder is 40 X — 2 = 16.2 lb. per sq. in. 

The total M. E. P. reduced to the low-pressure cylinder 
is, consequently, 12.05 -J- 16.2 -f- 16.5 = 41.75 lb. per sq. in. 
Area of low-pressure piston — .7854 X 66 2 = 3,421.2 sq. in. 

(a) Using formula 105, Art. 1495, 

_ PZ^ y_ 44.75x4x3,421.2x70x2 _ o 
LH - P --^000"- 3p00 -2,o98 HP. 

Ans. 

(b) The work done in each cylinder is proportional to the 
M. E. P. of the cylinder reduced to the low-pressure cylinder. 
Hence, the percentage of total work done in the high-pres- 
sure cylinder is .'„ = .269 = 26.9$. Ans. The percentage 

44. < o 

16.2 
of work done in the intermediate cylinder is , . ' = .362 = 

J 44. 7o 

36.2$. Ans. Lastly, the percentage of the work done in 

16 5 
the low-pressure cylinder is . * » = .369 = 36.9$. Ans. 

44. i d 

(829) See Arts. 1460 to 1463. 

(830) The volume discharged in 1 stroke is 

IS 2 X. 7854X24 . 

■ cu. it., and the volume discharged in 1 

1, < 28 



STEAM AND STEAM ENGINES. 291 

. 18 2 X .7854 X 24 X 175 X 2 „ . „ . 

minute is . Formula 103, Art. 

l,72o 

1395, gives the work in foot-pounds which is done in 1 
stroke; hence, if multiplied by the number of strokes per 
minute and divided by 33,000, the result will be the horse- 
power. The pressure is 62.4 lb. Substituting these values 
of the pressure and volume in formula 103 and dividing 
by 33,000, we have 
144X62.4X18 2 X. 7854X24X175X2 ' oor , 

1,728X33,000 = ' 3 ' 36 - 825 ^sepower. 

Ans. 

(831) Dividing the diagram into 10 equal parts, and 
measuring the ordinates drawn at the middle of these divi- 
sions, the sum of their lengths for diagram A is 7. 16", for 
diagram B, 7.24", and for diagrams CandD, 4.68" and 4. 72", 
respectively. Dividing these sums for A and B by 10 and 
multiplying by 60, the scale of spring, we have 

7 16 
for A, ^-— X 60 = 42.96 lb. per sq. in., 

7 24 
for i>, --— X 60 = 43.44 lb. per sq. in. 

Adding and dividing by 2, the mean effective pressure for 

.u u- u r a • t a * u 42.96 + 43.44 
the high-pressure cylinder is round to be ■ ^ • = 

2 

43.2 lb. per sq. in. 

The M. E. P. for the low-pressure cylinder is, 

for C, — — X 30 = 14.04 lb. per sq. in., 

a 79, 
for D, —-- X 30 = 14.16 lb. per sq. in. 

Adding and dividing by 2, the M. E. P. for the low-pres- 

. u.04+14.16 . . . „ 
sure cylinder is- — — =14.1 lb. per sq. in. 

2 

Reducing the M. E. P. of the high-pressure cylinder to the 

area or the low-pressure piston, we have 43.2 X t— ^ —- — = 

v F ' 20 2 X .7854 

18.252 lb. per sq. in. 
G. G, IV.— 18 



292 STEAM AND STEAM ENGINES. 

Hence, 14.1 + 18.252 = 32. 352 lb. per sq. in. = the M. E. P. 
if the work was all done in the low-pressure cylinder. 

PLAN 



Therefore, I. H. P. 



33,000 



32.352 XjfX 20 a x .7854 X 230 X 2 ._,_, 1 T ^ _. . 

33,000 - = m.lI.H.P. Ans. 

(832) Using a scale divided into 30ths of an inch, and 
locating for this case a point -If" above the vacuum line 
(which should be drawn previous to this), we draw through 
this point a line parallel to the atmospheric line. The length 
of the portion of this line included between the bounding 
lines of the diagram is 2.95", and the absolute pressure 
which it represents is 28 lb. The whole length of the dia- 
gram is %y. The weight of a cubic foot of steam at this 
pressure is, from the table of the Properties of Saturated 
Steam, .069545 lb. The M. E. P. was found to be 30.75 lb. 
per sq. in. in example 819. 

Using formula 107, Art. 1507, 

n 13,750 X 2.95 X .069545 _ 

Q = — 7T7nrz =-= = 26. 21 lb. per I. H. P. per hour. 

** 30.75 X 3.5 r - a 

Ans. 

(833) For (a) and (b) see Art. 1473. (c) No. The 

stroke may be so short that the piston speed will be low, 
although the rotative speed may be high. 

(834) This example is worked like the previous examples 
where the horsepower is to be found from the diagrams. 
Dividing the length of the diagrams into 10 equal parts, and 
measuring the middle ordinates, their sum for diagram A is 
found to be 6.04", and for B, 6.62". The mean ordinate for 

.604, and for B, ——- = .662. Hence, the total 

604 +.662 
~~ — X 60 = 37.98 lb. per sq. in. 

! H p = 87.98X1X18- X J854X300X2 = ^ R p ^ 

(835) Drawing a line parallel to the atmospheric line, 
and at a distance from the vacuum line corresponding to a 



A 


_ 6.04 

10 


M. 


E. P. 



STEAM AND STEAM ENGINES. 293 

pressure of 40 lb., absolute, the length included between the 
bounding lines of diagram A is 1.46", and the length between 
the bounding lines of diagram i? is 1.72". The length of 
both diagrams is 2.6". The weight of a cubic foot of steam, 
at an absolute pressure of 40 lb., is .097231 lb. The M. E. P. 
for card A = .604 X 60 = 36.24 lb., and for card B, .662 X 
60 = 39.72 lb. (See example 834.) 

Using formula 107, Art. 1507, we have, for card A, 

13,750 XI 46 X .097231 

* 36.24 X 2.6 

and for card B, 

_ 13,750 X 1.72 X. 097231 
U ~ 39.72X2.6 X 22.267 lb. 

Taking the average of both cards, 

n 20.716 + 22.267 rti (Al , _ „ „ , A 

Q = ■ = 21.49 lb. per I. H. P. per hour. Ans. 



(836) Formula 105, Art. 1495, gives 

_ PLAN _ 43.4 X || X 22 2 X .7854 X 200 _ 
" ' ' " : 33,000 ~~ 33,000 

150 H. P., nearly. Ans. 

(837) By formula 104, Art. 1457, the real cut-off 
in small cylinders is 

b - S —T-i - - ;375 +- 08 _ :i^ 
~l + t~ 1.08 "1.08" 

The ratio of expansion is, therefore, 

k .455 

The volumes of the large and small cylinders are propor- 
tional to the square of their diameters. 

V: v 

V 68 2 



294 



STEAM AND STEAM ENGINES. 



By formula 109, Art. 1527, the total ratio of expan- 

S1 ° n 1S n eV 2.374 X 68 2 i0 , 

£ = = — -5 =12.2. Ans. 

(838) (a) and (b) See Art. 1528. 

(c) All piston valves are balanced. 

(839) (a) 143 + 460 = 603° ; 256 + 460 = 716° ; 47.2 + 
460= 507.2°. 

(b) 785 - 460 = 325° ; 492 - 460 = 32° ; 443 - 460 = - 17°. 
That is, 17° below 0. 

(840) See Art. 1508. 

(841) As the cut-off becomes less and less, the com- 
pression is increased. The diagrams would be about as 
shown in Fig. 76. 




(a) 



(&) 




Fig. 76. 

The increase of compression is the distinguishing feature 
of the diagrams. 

(842) Length of stroke : length of card :: length of 
lever : distance from point, 

or 32 : 3£ :: 8 x 12 : x. 

8 X 12 X 31 ,_ . A 

x = — = 9f in. Ans, 

64/ 



STEAM AND STEAM ENGINES. 295 

(843) Assume the engine to be 16" X 20", and to make 
180 R. P. M. Suppose the M. E. P. to be 42.7 lb. per sq. in. 
Using formula 103, Art. 1395, 

f^-** = 33,000 ^ X^X 

2 = 156.1 H. P. Ans. 

Using formula 105, Art. 1495, 

PLAN _ 42.7 X-ff X 16 2 X .7854 X 180 X 2 _ rp 

33,000 ~ 33,000 " ^^ , F ' 

Ans. 

The student must assume different values from those 

given above when answering this question. 

14 7 

(844) (30 — 23.9) X -^— = 3 lb. = absolute pressure in 

the condenser. The total heat of a pound of steam at a 
pressure of 3 lb. above vacuum is, from the table of the 
Properties of Saturated Steam, 1,125.144 B. T. U. Applying 
formula 108, Art. 1520, 

w= //-,, + 32 = 1,125.144-124 + 32 = ^ ^ 
/j — / 2 105 — o2 

of water required per pound of steam. Hence, total weight 
of water required per minute = 

19.49X271.6X27.8 0/1 _ 1K A 

■ — — — = 2,452.6 lb. Ans. 

oO 



STEAM BOILERS. 

(QUESTIONS 845-894.) 



(845) See Arts. 1552 and 1553. 

(846) See Art. 1541. 

(847) See Arts. 1533, 1534, and 1540. 

(848) See Art. 1556. 

(849) See Art. 1547. 

(850) According to formula HO, Art. 1546, the air 

required for 1 lb. of fuel is 

A = 1.52 (6^+3 H) 

= 1. 52 (94 + 3 X . 5) = 145. 16 cu. ft. 
Then, 17 lb. would require 17 X 145.16 == 2,467.7 cu. ft. 

Ans. 

(851) Formula 111, Art. 1547, gives 

= 145 X 94 + 620 X .5 = 13,940 B. T. U. 

Ans. 

(852) See Arts. 1565 and 1567. 

(853) (a) Using the second rule in article 1547, 
we have 

13,675 +■ 966 = 14.16 lb. of water evaporated from and at 
212°. Ans. 

(b) 360 lb. of coal gives out 360 X 13,675 = 4,923,000 
B. T. U. Of these, 65 per cent, are used. 4,923,000 X M = 
3,199,950 B. T. U. absorbed by water. From the table of 
the Properties of Saturated Steam, the heat required to 
raise a pound of water from 85° and change it into steam at 
100 lb. pressure is 1,132 B. T. U„ nearly. Then the total 
water evaporated is 

3,199,950 ~- 1,132 = 2,826.8 lb. Ans. 

For notice of copyright, see page immediately following- the title page. 



208 



STEAM BOILERS. 



(854) Formula 113, Art. 1604, gives P 



2St e 

d ' m 



In this case, 5=11,000, t — 
Art. 1593, and d = 44. 

2 X 11,000 XfX 



Hence, P — 



44 



.75, from Table 31, 



= 140. Gib. Ans. 



(855) (a) HN0 3 . lx 1 



1 part hydrogen, ) by 
1 x 14 = 14 parts nitrogen, ) weight, 
3 x 16 = 48 parts oxygen. 

63 
The substance is then composed by weight of 

1.59$ hydrogen, 
22.22$ nitrogen, 



_i_ 

6 3 

JL.4 
6 3 

ff = 76.19$ oxygen. 



100.00 



(b) N 2 O s 



(') H 2 SO a . 



2X14 
3X 16 



2X 1 
1 X 32 
3 X 16 



28 
48 

76 

2 
32 

48 

82 



18 — 

7 

4 8 



|= 36.84$ nitrogen, 
63.16$ oxygen. 






2 

82" 

3.2 

82 



100.00 

2.44$ hydrogen, 
39.02$ sulphur, 
58.54$ oxygen. 



100.00 

(856) See Arts. 1557 and 1558. 

(857) See Art. 1560. 

(858) (a) See Art. 1543. 

(b) See Art. 1544. 

(c) One pound of carbon requires 5.8 lb. of air to burn it 
to C O gas. 8 lb. of carbon, therefore, requires 8 X 5.8 = 
46.4 lb. of air. Ans. 

(859) See Arts. 1598 to 1602, 

(860) See Art. 1575. 

(861) See Art. 1554. 



STEAM BOILERS. 299 

(862) See Arts. 1562, 1554,1555,1556,1557, 
and 1568. 

(863) See Arts. 1581 and 1583. 

(864) See Arts. 1549, 1550,and 1551. 

(865) The heat given up by coal is 320 X 13,450 B. T. U. 
The heat taken up by water is 2,500 X 966 B. T. U. 

Hence, the per cent, of heat utilized is 

2,500X966 _ w . 

aaox 13,45 = 5b - 1 ^ Ans - 

(866) (a) From 900 to 1,000 sq. ft. Ans. 

(b) Probably about 30 sq. ft. Ans. 

2 500 

(c) Actual horsepower = ' = 72.46. 

72.46-60 ._* A 

— — — = 20.77$. Ans. 

60 

(867) See Arts. 1601, 1602, and 1562. 

(868) See Arts. 1548 and 1551. 

(869) (a) Using formula 111, Art. 1547, 

B = 145 C+ 620/7 = 145 X 84.7 + 620 X 13.1 = 20,403.5 
B. T. U. Ans. 

(b) By the second rule in Art. 1547, 

20,403.5 -T- 966 = 21.12 lb. of water. Ans. 

(c) By formula HO, Art. 1546, 

A = 1.52 (C + 3 H) =1.52(84.7 + 3 X 13.1) -=188.5 cu. ft. 

Ans. 

(870) Formula 113, Art. 1 604, gives P = ^X 

In this case, 5=9,000, t = T \", * = .60, from Table 31, 
Art. 1593, and d— 60'. 

u ax 9,000 x -A- x. 60 lvau . . 

P= ' n^ = ^9 lb., nearly. Ans. 

60 

(871) By formula 113, Art. 1604, 

25/r 2 X 9,000 X A X 1 ' .. , 

F-=- — j — = : =-=- = 703 lb. Ans. 



300 STEAM BOILERS. 

(872) See Arts. 1589 to 1592 and Art. 1597. 

(873) See Arts. 1588 and 1596. 

(874) See Arts. 1615 and 1616. 

(875) The number of B. T. U. absorbed per hour by 
water is 280 X 14,000 X .70 = 2,744,000. One horsepower = 
33,330 B. T. U. Hence, the H. P. of boiler is 2,744,000 -f- 
33,330 = 82.3 H. P. Ans. 

(876) (a) See Art. 1 608. 

(/;) In the boiler shown in Fig. 432, Art. 1561, the heat- 
ing surface includes the inner surface of the furnace flues, 
the inner surface of the tubes, and the portions of the heads 
allowed to come in contact with the gases. 



(b) Coal burned per hour = Lo * 2, ° — = 300 lb. 



(877) (a) See Art. 1605. 

(b) Coal but 
Grate area = 5J X H- 
Rate of combustion = — —=15.6 lb. per sq.ft. per 

O2" X 0~2 

hour, nearly. Ans. 

(878) The higher the rate of combustion, the less the 
quantity of air required per pound of coal. 

(879) (a) 175,579 -4- 21,400 = 8.2 lb. Ans. 

(b) From Table 32, Art. 1617, the factor of evapora- 
tion for 36° and 60 lb. pressure is 1.213. 

8.2 X 1.213 = 9.947 = equivalent evaporation from and 
at 212°. Ans. 

(c) Lb. of water per hour = W= — £- — . 

F, the factor of evaporation, = 1.213. 
Then, by formula 114, Art. 1618, 

175 ' g79 XlS13 

IV F 99 

HP =-w= — m — =28a6 - Ans - 

(880) See Art. 1541. 



STEAM BOILERS. 301 

(881) To determine the safe working pressure, use 
formula 113, Art. 1604. 

n %Ste 2 x 9,000 X -A- X .64 OA „ 

P— = — ±= — ±-5 — 80 lb. per sq. in. 

a 4o 

The boiler would, therefore, be unsafe. Ans. 

(882) Using formula 112, Art. 1603, F=Pdl = 

120 X 51 X 11 X 12 = 1,028,160 lb. Ans. 

(883) The answer to this question is left to the stu- 
dent's discretion. See Arts. 1605, 1607, 1614, and 
1616. 

(884) 1 ' 1WX Q ig X:68 = 291.7H.P. An, 

DO, oO\J 

/ook\ , N 1,100 X 13,000 X ;68 1AACC oiu * 

(885) (a) — ~- = 10,066.3 lb. of water. 

A 

Ans. 
(b) 1,100 -=- 966 = 1.14 lb., nearly. Ans. 

(886) See Arts. 1573, 1571, and 1572. 

(887) The rule given in Art. 1609 is used in this case. 
Heating surface in shell = 

48 X 3.1416 X 138 ( = 11'- 6") X f = 13,873.3 sq. in. 
Heating surface in tubes = 

62 X 138 X U X 3.1416 =' 67,198.8 sq. in. 
Area of head = 48 2 X .7854 = 1,809.6 sq. in. 
Combined area of tubes = 2.5 2 X .7854 X 62 = 304.3 sq. in. 
Then, according to the rule, heating surface = 
13,873.3 + 67,198.8 -f 2 X | X 1,809.6 - 2 X 304.3 _ 
144 
575.5 sq. ft. Ans. 

(888) According to formula 114, Art. 1618, 

H WF 
H= 34.T 

In this case, F (see Table 32) is 1.2035. 

„ rr 12,370 X 1.2035 _, 

Hence, H = — — = 431.5 H. P. Ans. 

34.5 



302 STEAM BOILERS. 

(889) B. T. U. taken by water = 5.8 X 966. 

B. T. U. given by coal = 12,750. 

t?«: • 5.8 X 966 ._ ... . 

Efficiency = —————= 43.94$. Ans. 
1/w, 750 

(890) See Arts. 1556, 1554, and 1560. 

(891) Since the product is S0 2 , the composition of this 
product by weight is 

S = 1 X 32 = 32 
O = 2 X 16 = 32, 



or 50$ of each. Hence, 1 lb. of sulphur requires a pound of 
oxygen. The scheme is, therefore, as follows: 

1 lb. sulphur sulphur ...1 lb. ) rt ., „ 

( i iu P lb. S0 2 

a ok i b air j oxygen... 1 lb J 

^° ( nitrogen ... 3. 35 lb. 3. 35 lb. nitrogen. 

5.35 5.35 

(892) See Arts. 1577 and 1569. 

(893) =r^ = 27.14 sq.ft. Ans. 

(894) (a) From Table 32, Art. 1617, the factor of 
evaporation is 1.211. 800 X 1.211 = 968.8 lb. Ans. 

(b) From formula 114, Art. 1618, 






STEAM BOILERS. 

(QUESTIONS 895-945.) 



(895) See Arts. 1619 to 1626. 

(896) By formula 115, Art. 1621, W = pA. 

A = .7854 X (2^) 2 = 4.91 sq. in. 
Then, W = pA = 60 X 4.91 = 294.6 lb. Ans. 

(897) See Art. 1624. 

(898) Use formula 117, Art. 1624 9 in this case. 



d AlA-p± 



(a) 



^(BgX 10-84) x8^ = - Ans ^ 



ss 



rf= (75xl0-24)x3i =28 ^ Ans 

oo 



</= (8oxio-a4)x3 t = 3a86 , Ans _ 

oo 

f ,v , (90 X 10-24) X 3| _. ,„ , 
(^/) d—~ — '- ± = 34.84". Ans. 

oo 

(899) See Art. 1626. 

(900) See Arts. 1679 and 1680. 

(901) Formula 118, Art. 1627, should be used. 
A - - 5S ~ • 5X6 ° ° - 2 7<>7 so in 

A -7+T0 - 100T10 ~ 2 ' 72 ' q - • 



/2 7*27 
The diameter corresponding to this area is \ ' = 1^", 

nearly. Ans. 

For notice of copyright, see page immediately following the title page, 



304 



STEAM BOILERS. 



(902) Formula 118, Art. 1627, gives 
.dS .5X2,500 



A 



/ + 10 80+10 



14 sq. in., nearly. 



Then, diameter 



= / 



14 



= 4.22 in., say 4^ in. Ans. 



.7854 

(903) See Art. 1628. 

(904) See Arts. 1629 to 1633. 

(905) The student may receive suggestions from Arts. 
1677 and 1678. 

(906) See Arts. 1635 and 1637. 

(907) 110 lb., gauge = 124.7 lb., absolute. From the 
table of the Properties of Saturated Steam the total heat 
of a pound of steam above 32°, at 124. 71b. pressure, is 1,186.8 
B. T. U. Then, 1,186.8 - (50 - 32) = 1,168.8 B. T. U. = 
number of heat units necessary to change a pound of water 
at 50° into steam of 110 lb. pressure, gauge. Number of 
heat units saved by heating the feed-water = 206 — 50 = 156 
B. T. U. Hence, the per cent, gained = 156 ~ 1,168.8 = 
13.350. Ans. 

(908) See Arts. 1638 and 1639. 

(909) See Arts. 1641 and 1642. 

(91 0) See Art. 1646. 

(911) See Arts. 1645 and 1647. 

(912) See Art. 1648. 

(913) See Arts. 1650 to 1652. 

(914) See Arts. 1654 and 1657. 

(915) See Arts. 1656 and 1658. 

(916) See Art. 1660. 

(917) Using Table 33, Art. 1663, the horsepower is 
found, in the column headed 125 and opposite 72, to be 934 
H. P. Ans. 






STEAM BOILERS. 



305 



(918) Using formula 119, Art. 1662, 

7.6 7.9 



/ 



= 135 (- 



60/ 



.783" of water. Ans. 



60 + 460 440 + 460, 

See Art. 1665. 

See Arts. 1667 and 1668. 

See Arts. 1669 and 1662. 

See Arts. 1670 and 1671. 

See Art. 1673. 

See Art. 1674. 

See Arts. 1675 to 1678. 

See Arts. 1683 to 1686. 

See Art. 1687. 

See Arts. 1688 and 1689. 

See Arts. 1690 to 1694. 

See Arts. 1695 to 1697. 

See Art. 1699. 

See Art. 1698. 

See Arts. 1700, 1701, and 1702. 

See Art. 1 703. 

See Art. 1705. 

See Art. 1706. 

See Art. 1707. 

See Art. 1711, Section IX. 

See Arts. 1713 and 1714. 

Using formula 1 20, Art. 1714, and obtaining 
L ='881.4, nearly, and t = 331°, from the table of the 
Properties of Saturated Steam, we have 

Q = gg^j [^ (135-44) - (331-135)1 = .9508 = 95.08*. 

Ans. 



(919 
(920 
(921 
(922 
(923 
(924 
(925 
(926 
(927 
(928 
(929 
(930 
(931 
(932 
(933 
(934 
(935 
(936 
(937 
(938 
(939 
(940 



306 STEAM BOILERS. 

(941) See Art. 1719. 

(942) See Arts. 1 7 1 7 and 1 7 1 8. 

(943) See Arts. 1721 to 1724. 

(944) (a) Actual quantity of water evaporated = 
114,962.7 X .98 = 112,603.5 lb. 

Percentage of combustible = 100 — 4.81 = 95.19$. 

Combustible = 12,000 X .9519 = 11,422.8 lb. 

Factor of evaporation, from Table 32, = 1.1138. 

Water evaporated per pound of coal, actual conditions, = 
112,663.5 4- 12,000 = 9.39 lb. Ans. 

Water evaporated per pound of combustible, actual con- 
ditions, = 112,663.5 -f- 11,422.8 = 9.86 lb. Ans. 

(b) Water evaporated per pound of coal from and at 
212° = 9.389 X 1.1138 = 10.46 lb. Ans. 

Water evaporated per pound of combustible from and at 
212° = 9.863 X 1.1138 = 10.98 lb. Ans. 

(c) Equivalent evaporation from and at 212° = 112, 663. 5 X 
1.1138 = 125,484.6 lb., or 12,548.46 lb. per hour. 

The horsepower developed is, therefore, 12,548.46 -=- 34£ = 
364 H. P., nearly. Ans. 

(945) Water used per minute = ■— -£ — = . 9333 cu. ft. 

v ' 62.5 X 60 

9333 
Water displaced per stroke = ' X 1,728 = 80.6 cu, 

20 

in. = volume of cylinder. 

A cylinder 4J" X 5", or 4" X 6J", would answer. 






WATER-WHEELS. 

(QUESTIONS 946-998.) 



(946) Applying formula 122, Art. 1730, 
H >R= 21X 62.5X14 = 88 ; 4 Ang 

5o0 

(947) The quantity of water that will be available for 
the 10 working hours is 21 X f-jj- = 50.4 cubic feet per 
second ; therefore, the theoretical horsepower will be 

50 - 4 * 5 y Xl4 =80 A H.P. 

Applying rule I, Art. 1737, the available work is 
80 T 2 T X .75 = 60.136 H. P. Ans. 

(948) Applying formula 123, Art. 1731, 

K=c 2 lV/i = .9S 2 Xl$ X 62.5X75 = 81,033.75 foot-pounds 
per second. Ans. 

(949) Applying formula 126, Art. 1732, 

v 23 

P=z IV- = 7 X 62.5 X 77--T77 = 312.888 pounds. Ans. 
g 32.16 r 

(950) The weight of water flowing in the jet is 
W— .25 X 38 X 62.5 = 593.75 pounds per second. 

Applying formula 129, Art. 1735, and remembering 

that, since the jet is perpendicular to the surface, a — 90° 
and cos a = 0, 



P= (1 - cos a) Wll - ^\ - 



8- 10 
32.16 



593.75 X ^ X tt!^- = 380.91 pounds. Ans. 
19 o2. 10 

For notice of copyright, see page immediately following the title page. 

q. a. TV.— 10 



308 WATER-WHEELS, 

(951) Applying formula 128, Art. 1734, and sub- 
stituting, 

P = (1-costf) W- = (1-.866) X 593.75 X J^t; = 94.01 
v g • 32.16 

pounds. Ans. 

(952) In accordance with the principles of Hydro- 
mechanics, the head that would produce a pressure of 

125 
125 pounds per square inch is h = feet ; and the velocity 

of flow from the nozzle is 



.98|/2^7/ = .98X8.02 y- 



125 



133.385 feet per second. 
.434 x 

In accordance with the principles developed in Art. 1735, 

the water will leave a hemispherical cup with no absolute 

velocity when the velocity of the cup is equal to one-half 

the velocity of the jet; hence, the velocity of the cup must 

be 133.385 -~ % = 66.692 feet per second. Ans. 

(953) From formula 130, Art. 1735, the pressure is 

P= .0622 — (v - v'Y = .0622 l*™'* (133.385 - 66.692) 2 = 

V loo. 00O 

324.089 pounds, 

and from formula 121, Art. 1727, the energy is 

K—Pv' = 324.089 X 66.692 = 21,614.1 foot-pounds per 
second. Ans. 

(954) The theoretical horsepower in the water is, from 
formula 122, Art. 1730, 

36X625X23 = 94()91H p; 

00O 

Hence, according to the definition of efficiency given in 
Art. 1737, the efficiency is 

62 J ~ 94.091 = .66425 = 66.425 per cent. Ans. 

(955) The theoretical horsepower is 

8-W X6a-6X 482 = lsft685H , P. 

00O 



WATER-WHEELS. 309 

The theoretical power in the water due to the pressure at 

the nozzle is the same as the theoretical power that would 

be developed if the water were to fall freely through a 

height equal to the head required to produce the given 

pressure. Since a head of 1 foot produces a pressure of 

.434 pound per square inch, the head that would produce a 

192 
pressure of 192 pounds per square inch is . , and the 

theoretical horsepower due to the pressure is 

2.75 X 62.5 X ~r 
434 

— '-^ = 138.241 H. P. 

550 

(a) Since the wheel develops 92 horsepower, it uses the 
water delivered to the nozzle with an efficiency of 
92 



138.241 



.6655 = 66.55 per cent. Ans. 



(b) The efficiency of the whole plant is equal to the 
horsepower actually developed, divided by the total theoreti- 
cal horsepower = 

92 
— -— - = .61079 = 61.079 per cent. Ans. 
150.625 

(956) The total theoretical power in the water used is 

78 X 62.5 X 7.5 



550 



= 66.476 H. P. 



(a) The power lost through the resistances in the pipe is 
equal to the amount of power that would be developed if 
the water were to fall freely through a distance equal to the 
loss in head, or 

78 X 62.5 X. 32 . QQa „ _ . 

— = 2.836 H. P. Ans. 

550 

(b) The loss in efficiency due to resistances in the pipe is 
equal to the loss in power due to those resistances divided 
by the total theoretical power = 

2.836 



66.476 



; = .04267 = 4.267 per cent. Ans. 



310 WATER-WHEELS. 

(957) The theoretical weight W of water required per 
second to furnish 75 horsepower with a fall of 40 feet is equal 
to the total number of foot-pounds of work that must be 
done in one second, divided by 40, the distance in feet 
through which the water falls ; therefore, 

W = ■ — — — 1,031.25 pounds per second; 

and, from rule II, Art. 1737, the total weight required is 
1,031.25 



.68 
1,516.5 



1,516.5 pounds. 
24.264 cu. ft. Ans. 



62.5 

(958) {a) The velocity of entrance, from the conditions 
of the problem, is 

v e = 1^ x 6 = 9 feet per second ; 

therefore, applying formula 132, Art. 1743, 

h = l.l^- = 1.1 X 77^= 1-385 feet. Ans. 
%g 64.32 

(b) The diameter D of the wheel (see Art. 1743) is 
equal to the total head H minus the head // and the clear- 
ance between the wheel and trough; i. e., 

D = 23 - (1.385 + .125) = 21.49 feet, 

or, in even numbers, 21-| feet. Ans. 

(c) Applying formula 133, and substituting, 

AT= 19.1-^=19.1 X^t = 5.33. Ans. 
JJ 21.5 

(d) From formula 134, Z= 2-J- D to 3 D\ choosing the 
former value, 

Z = 2i D = 2% X 21. 5 = 53. 75, say 54. Ans. 

(e) Assuming a mean value for the coefficient in formula 
136, and substituting, 

£-3.5_2=3.5 X -^-= 1.454 feet = 17 L say 18 inches. 
dv lXb Ans. 



WATER-WHEELS. 311 

(959) (a) See Arts. 1741 and 1744 to 1747. 
(b) See Arts. 1751 and 1752. 

(960) Since the velocity of the circumference of the 
wheel should be one-half the velocity of the current (see 
Art. 1753), the velocity of the circumference is 

v = -§- — 4^ feet per second; 
and, from formula 133, Art. 1743, 

N =■ 19.1 -jz= 19.1 X -~ = 6.1 revolutions per minute, 
say 6 per minute. Ans. 

(961) The quantity of water Q flowing in the stream is 
Q = av=6xliX 10-J- = 94J- cubic feet per second ; 

hence, assuming the velocity of the circumference of the 
wheel to be equal to one-half of the velocity of the current, 
and applying formula 137, Art. 1754, 

H.P. = . 0012 (v — y t ) v y (2 = .0012XpL0i-.5£)x 5^X94^ = 3.1. 

Ans. 

(962) (a) and (b) See Art. 1763. 

(963) The velocity with which the water flows from 
the nozzle is 

z/=.98x |/2 < ^//=.98x8.02| / — — =106.71 feet per second; 

and, since the maximum efficiency is obtained when the 
circumferential velocity of the wheel equals one-half the 
velocity of the jet (see Art. 1764), the velocity of the cir- 
cumference is 106.71 v2 = 53.36 feet per second. Hence, 
applying formula 133, Art. 1743, 

v 53 36 

N = 19. 1 -= = 19. 1 X — ^— = 509. 5 rev. per min. Ans. 

(964) The velocity with which the water flows from 
the nozzle is 

v= .98 4/2 £-/; = .98 X 8.02 4/285 = 132.685 feet per second; 
hence, as in the last example, the velocity of the cups 
should be 

132.685 -r2 = 66.342 feet per second. 



312 WATER-WHEELS. 

Applying formula 133, Art. 1743, 

N—19.1~ 1 orD= 19.1-^, we have 

D A' 

P C Q A 

D= 19.1 X n " = 5.06 feet, say 5 feet. Ans. 
2o0 

(965) (a) and (//) See Arts. 1770 and 1771. 

(966) Because, when the water leaves the wheel with a 
considerable absolute velocity, it carries a considerable 
quantity of energy with it which has not been utilized in 
doing work. (Art. 1731.) 

(967) See Art. 1762. 

(968) See Art. 1769. 

(969) See Art. 1772. 

(970) See Art. 1773. 

(971) {a) See Arts. 1792 and 1793. 
(b) See Arts. 1790 and 1791. 

(972) Applying formula 149, Art. 1796, 

33 

e^= -y^ — = ■ — = .48134 feet = of inches. 

1 *■ V * 27 X ^ X 19i Ans. 

(973) (a) and (b) See Art. 1799. 

(974) (a) See Art. 1741. 

(b) See Art. 1748. 

(c) See Art. 1756. 

(975) Applying formula 149, Art. 1796, 

e % = -y-^ = — — = .5337 feet = 6^- inches. 

-*.*.*. 28x £_|£ x21 Ans . 

(976) Applying formula 133, Art. 1743, 

71 1 Q K 

Z>=19.1 X -r- r = 19.1 X ~r = 2.979 feet, 
N 125 



WATER-WHEELS. 313 

from which r = \ D= 1. 480, or, in even numbers, r=l% feet = 
18 inches. Ans. 

(977) See Art. 1803. 

(978) The head required to produce the velocity with 
which the water leaves the draft tube is 

/ ' = ^=64^ = 1 - 5547feet; 

therefore, since the energy in the water is directly propor- 
tional to the head (see Art. 1727), the proportion of the 
total energy lost in producing the velocity of discharge is 
1.5547 -f- 18 = .08637 = 8.637 per cent. Ans. 

(979) See Art. 1804. 

(980) The loss in head is 12 feet - 8£ feet = 3£ feet ; 
hence, since the available energy is directly proportional to 
the head (see Art. 1727), the loss is 

3£ -~ 12 = . 29167 = 29. 167 per cent. 

of the original energy, and there is a consequent loss in 
efficiency of 29.167 per cent. Ans. 

(981) (a) and (b) See Art. 1809. 

(982) See Art. 1809. 

(983) See Art. 1768. 

(984) {a) See Arts, 1812, 1815, and 1818. 
(b) See Art. 1812. 

(985) See Art. 1813. 

(986) See Art. 1814. 

(987) A = ^- = ^ = 37.5sq. ft. Ans. 

(988) See Art. 1794. 

(989) See Art. 1832. 

(990) The theoretical weight J7 r of water required per 
second to furnish 125 horsepower with a head of 32 feet is 



314 WATER-WHEELS. 

equal to the total number of foot-pounds of work that must 
be done in one second, divided by 32, the distance in feet 
through which the water falls; therefore, 

. W= — = 2,148.437 pounds per second; 

O/i 

and, from rule II, Art. 1737, the total weight required is 
2,148.437 



70 



= 3,069.196 pounds per second. 



3,069.196 , ni . , A 

— = 49.1 cu. ft. per second. Ans. 
62. 5 

(991) Since the average flow per second during the 
summer is 3^- cubic feet, the water that will be furnished by 
the flow of the stream, if used during 10 hours out of the 24, 
will be 3^ X ff = 8.4 cubic feet per second. 

The supply from the reservoir will furnish 

22,000,000 a nAn "... , 

— ^- — -— — = 6.642 cubic feet per second 

92 X 10 X 60 X 60 v 

for an average of 10 hours per day during the 92 days of 
the summer ; hence, the total available flow is 8. 4 -j- 6. 642 = 
15.042, say 15 cu. ft. per second, and the horsepower with 
an .efficiency of .68 per cent, is 

15 X 62.5 X 25 ^ gg = ^ R p n Afts 

550 

(992) (a) See Art. 1741. 

(b) See Art. 1756. 

(c) See Art. 1759. 

(d) See Art. 1812. 

(e) See Art. 1770. 

(993) (a) See Arts. 1759 to 1763. 
(b) See Arts. 1767 to 1771. 

(994) Applying formula 138, Art. 1755, 

H. P. =.001 (v — v x ) vv i F = 

.001 X (12 - 6) X 12 X 6 X (5 X H) = 3.24. Ans. 



WATER-WHEELS. 315 

(995) The velocity of flow from the nozzle is 



v= .99i/2gA= .99 X 8.02 X i/750 = 217.44 feet per second 
therefore, the area of the nozzle must be 

A = Q = °;° it = .016096 sq. feet, or .016096 X 144 = 
v 217.44 n 

2.3178 square inches. 
The diameter corresponding to this area is 



d= V-^i = y—f^l = L7179 inches = Iff inches. Ans. 

(996) The head equivalent to the absolute velocity 
with which the water leaves the wheel is 

*'=.fe=6& = - 996£ebt ' 

and the head lost in the fall from the wheel to the tail water 
is 10 inches = .833 feet, making a total loss in head, when 
the water is discharged from the wheel, of .995 -\- .833 = 
1.828 feet. Since the loss in energy is proportional to the 
loss in head, the percentage of the original energy lost is 

1 82S 

— — = .04062 = 4.062 per cent. Ans. 
45 r 

(997) The area of the draft tube must be 



A = — = — - = 20 square feet; 
v . 4 



and the diameter 



|/-^_ = / 3 



or, in even numbers, 5 feet. Ans. 



(998) The outflow area from the guide vanes must be 

28 
A = — = — - = .8485 square feet. 
v e do 

Applying formula 147, Art. 1795, 

A ■ A 

e ~ Zx - Z x s~ 28 X ] 
or 3 inches, nearly. Ans. 



A .8485 X 144 , .„. . . 

€ = ZYZrz-s = 28 X 1.625- 22 X A = * ™ ^ ' 



HYDRAULIC MACHINERY 

(QUESTIONS 999-1058.) 



(999) See Art. 1837. 

(1000) See Art. 1837. 

(1001) See Arts. 1839 and 1848. 

(1002) (a) See Art. 1841. 

(b) See Art. 1842. 

(c) See Art. 1 862. 

(1003) See Art. 1852. 

(1004) (a) See Art. 1849. 

(b) See Art. 1842. 

(c) See Art. 1861. 

(d) See Art. 1863. 

(1005) See Art. 1865. 

(1006) See Art. 1852. 

(1007) See Art. 1858. 

(1008) See Arts. 1870 and 1871. 

(1009) See Art. 1883. 

(1010) See Art. 1888. 

(1011) See Art. 1890. 

(1012) Since the pump is to work under ordinary 
conditions, we may make the volume of the air-chamber 
about three times the piston displacement for a single 
stroke. (See Art. 1885.) From the rule given in Art. 
1905, the displacement for a single stroke is 22 X .7854 X 
8 a X 1 = 1,105.843 cu. in.; therefore, the size of the air- 

For notice of copyright, see page immediately following the title page. 



318 HYDRAULIC MACHINERY. 

chamber will be 1,105.843 X 3 = 3,317.529 cu. in. = 2 cu. ft., 
nearly. Ans. 

(1013) See Art. 1890. 

(1014) See Art. 1891. 

(1015) See Art. 1897. 

(1016) See Art. 1898. 

(1017) See Arts. 1899 to 1902. 

(1018) See Arts. 1903 and 1904. 

(1019) Applying rule, Art. 1905, the displacement 
is found to be 

H X 2 X 75 X ' * = 69.813 cu. ft. per min. Ans. 

(1020) The displacement in gallons per minute is 
69.813 x 7.48 = 522.2; therefore, using rule in Art. 1909, 

the slip is ° 22 ; 2 ~ 4 — = .1383 = 13.83 per cent. Ans. 

(1021) See Art. 1909. 

(1022) In accordance with the definition given in 
Art. 1910, the useful work is 

325 X 8J X 280 = 758, 333 J ft. -lb. per min. Ans. 

(1023) The number of gallons discharged per minute is 

' , and the plunger speed is 7 X 10 = 70 feet per minute; 

therefore, applying formula 152, Art. 1916, and substi- 
tuting, we have 









d= 5.535 \' ^ = 5.535 V ^^L = 7 - G39 in -= '1 in -> nearly. 

Ans. 

(1024) (a) Applying formula 152, Art. 1916, and 
substituting, the diameter of the plunger is 

d— 5.535 y-^r = 5.535 y^— = 15. 158 in. = loin., say. Ans. 

O J.UU 



HYDRAULIC MACHINERY. 319 

(b) Applying formula 158, Art. 1921, and substituting, 
the size of the suction-pipe is 

d x - .35 4/£ = .35 4/750 = 9.585 in.; 

therefore, make the suction-pipe 10 in. Ans. 

(c) Applying formula 159, Art. 1921, and substituting, 
the diameter of the delivery-pipe is 

d i = .25 */G = . 25 4/750 = 6.847 in.; 

therefore, make the delivery-pipe 7 in. Ans. 

(1025) (a) 200 gallons per minute = discharge for both 
sides of pump, 200 -=- % = 100 gallons for each pump. Apply- 
ing formula 152, 

d = 5.535 \ % = 5.535 y ]°~= U in. Ans. 
S r 150 2 

(b) Applying formula 158, Art. 1921, and substituting, 

d 1 = . ( doi/U= .35/200 = 4.95 in.; 
therefore, make d 1 = 5 in. Ans. 

(c) Applying formula 1 59, Art. 1921, and substituting, 
d 2 = .25 4/^= .25 4/200= 3.536 = 3| in., say. Ans. 

(d) Applying formula 154, Art. 1918, and substitu- 
ting, 

H= .00038 Gh = . 00038 X 200 X 250= 19 H. P. Ans. 

(1026) (a) The number of gallons that will be dis- 
charged per minute by one pump is found by applying 
formula 153, Art. 1917. Substituting in the formula, 
G = .03264 d* S = .03264 X 14 2 X 100 = 639.744 gal. ; the 
discharge per minute for the two pumps is 639.744 X 2 = 
1,279.488 gal. ; and the discharge per hour is 1,279.488 X 60 = 
76,769.28 gal. Ans. 

(b) The horsepower of the pump is equal to the quotient 
obtained by dividing the product of the area of the steam 
cylinders, the steam pressure, and the plunger speed in feet 
per minute by 33,000, or 

„ 380.13X2X45X100 
ff= ~ ^000~ - = 103.67; 



320 HYDRAULIC MACHINERY. 

therefore, substituting in formula 155, Art. "1919, 

/;_ H - 103.67 - 213ft Ans 

" .00038 G ~ .00038 X 1,279.488 " * L6 

(1027) (a) The theoretical horsepower is equal to the 
number of foot-pounds of useful work done in one minute 
divided by 33,000, or 

80,000X8^X420 

33,000 X 60 X 4i,4i • 

(#) Applying formula 1 54, Art. 1918, and substituting, 

H= .00038 Gh = .00038 X ^— X 420 = 212.8 H. P. Ans. 

oO 

(1028) Applying formula 160, Art. 1922, and sub- 
stituting, 

D = ^ 5g A = 835.5 X 30,000 X 290 = ^^ ft ^ 

Ans. 

(1029) (a) Applying formula 153, Art. 1917, and 
substituting, 

C7 = .03264 d* S= .03264 X 15 2 X 100 = 734.4 gal. per min. 

Ans. 

(b) First find the required horsepower. By substituting 
the number of gallons just obtained in formula 154, Art. 

1918, 

H = .00038 X 734.4 X 310 = 86.51 H. P. 

Now substituting in formula 157, Art. 1920, we have 
for the diameter of the steam cylinder, 

therefore, make the diameter of the steam cylinder 27 inches. 

Ans. 

(1030) (a) Applying formula 153, Art. 1917, and 
substituting, 

G = . 03264 d* S = .03264 X ll 2 X 100 = 394.944 gal. per 
min. = 394.944 X 60 = 23,696.64 gal. per hr. Ans. 



HYDRAULIC MACHINERY. 321 

(b) Applying formula 1 54, Art. 1918, and substituting, 

H- .00038 Gh = .00038 X 394.944 X 300 = 45.024 H. P. 

Ans. 

(c) Applying formula 157, Art. 1920, and substituting, 



_ j/ 42,016.8 X H _ ,/ 42,016.8 X 45.024 _ 
r P X 5 " r ' 50 X 100 

19.45, say 19-J- in. Ans. 

(1031) First find the theoretical horsepower, 

18,000 X 8* X 225 _ 170 ^ 
H - R - 33,000 X 60 ~ 17 ' 045 ' 

The actual horsepower required is 17. 045 X 1. 5 = 25. 568 H. P. 
By applying formula 157, Art. 1920, we get for the 

diameter of the steam cylinder, 

, 4 /42,016.8 X 25.568 1 Q _„ _ 1 . . . 

* 50 x 110 = 13 - 976 ' or U in -' sa ^ Ans ' 

The diameter of the water plunger is found by substi- 
tuting in formula 152, Art. 1916, 



d= 5.535^ ^ = 5.535 V n ]*'°° l % := 9.138, or 9^ in., say. 
f S f 60 X 110 a 

Ans. 

The length of stroke may be taken as 12 inches, in accord- 
ance with good practice. 

To find the diameter of the suction-pipe, apply formula 
158, Art. 1921; whence, 



d x = .35 y 18, 6 ( | ) 0Q = 6.062 = 6 in., say. Ans. 
For the discharge-pipe, from formula 159, same article, 



< = . 25t / l» = 4.33in.; 



60 
therefore, make the diameter 4|- in. Ans. 

(1032) The total work done is 

W— 218 X 3,240,000 X 8J = 5,886,000,000 ft.-lb. ; 



322 HYDRAULIC MACHINERY. 

therefore, applying formula 162, Art. 1924, and substi 
tuting, 

87,000,000 ft. -lb. Ans. 

(1033) The area of the plunger is 
.7854 X 14 2 



144 



= 1.07 square feet, nearly. 



Since there are two pumps, the displacement, according to 
rule, Art. 1905, is 1.07x100x2 = 214 cubic feet per 
minute. The discharge is 

75,000 1fl „ 11Q .. f ' • , 

167.112 cubic feet per minute; 



60 X 7.48 



therefore, the slip, in accordance with rule given in Art. 
1909, is 

214- 167.112 



214 



= .2191 = 21.91 per cent. Ans. 



(1034) Applying formula 160, Art. 1922, and sub- 
stituting, 

n 835.5 Gh 835.5X4,000,000X125 KK OOQ aan ,, ., 
D=— w —= ^- _ = 55,998,660 ft -Ih. 

Ans. 

(1035) See Art. 1862. 

(1036) {a) The number of gallons discharged per 
minute is — '— — = 1,166-f; therefore, applying formula 158, 
Art. 1921, and substituting, 



d x — . 35 \/G = . 35 4/1,166! = 11. 955, say 12 in. Ans. 

{b) Applying formula 159, Art. 1921, and substitu- 
ting, 

< = .25|/l7 = .25|/1,166|= 8.539, say 8| in. Ans. 

(1037) Applying formula 154, Art. 1918, and sub- 
stituting, 

#"= .00038 Gh = .00038 X l^M^ x 480 = 304 H. P. Ans. 

60 



HYDRAULIC MACHINERY. 323 

(1038) (a) and (b) See Art. 1921. 

(c) See Art. 1877. 

(d) See Art. 1920. 

(1039) Applying formula 153, Art. 1917, and sub- 
stituting, 

G= .032<5±d 2 S= .03264 X 15 2 X 95 = 697. 08 gal. per min. 
= 697.68 X 60 = 41,860.8 gal. per hr. Ans. 

(1040) (a) The area of an 18-inch cylinder is 

.7854 Xi y ft 

144 n 

The total plunger displacement of the two pumps for the 
10 hours was 

1.767 X 2 X 2 X H X 12,480 = 220,521.6 cu. ft. Ans. 

(b) The plunger displacement in gallons for the 10 hours 
was 220,521.6 X 7.48 = 1,649,502 gallons, nearly; therefore, 
the slip was 

1,649,502-1,575,521 



1,649,502 



= .04485 = 4.485 per cent. Ans. 



(6') The pressure corresponding to the vacuum in the 

9.25 
suction-pipe is p = 5-7^7=- = 4.54 pounds; and the pressure 

corresponding to the difference in level of the centers of the 
vacuum gauge and the pressure gauge is s — 10.75 X .434 = 
4.666 pounds. 

Applying formula 161, Art. 1924, and substituting, 
the total number of foot-pounds of work is 
W=2 [1. 767 X 144 X (108+4. 54+4. 666) X 2. 5 X 2x12,480] = 
3,721,889,469.5 foot-pounds. 

Applying formula 162, Art. 1924, and substituting, 

D = jj x 1,000,000 = 3 ^^qqq' 5 x 1,000,000 = 

95,000,000 ft. -lb. nearly. Ans. 

(1041) (a) See Art. 1925. 

\b) See Art. 1932. 
(c) See Art. 1956. 
G. G. IV.— 20 



324 HYDRAULIC MACHINERY. 

(1042) (a) See Art. 1938. 
(b) See Art. 1942, 

(1043) See Art. 1944. 

(1044) See Art. 1946. 

(1045) See Art. 1945. 

(1046) See Art. 1966. 

(1047) Applying formula 163, Art. 1969, and 
remembering that since the ram is horizontal its weight 
does not affect the pressure, 

^•7854^(l-4) = .7854x8'x3,500(l-y) = 

171,531.36 lb. Ans. 

(1048) See Art. 1939. 

(1049) Applying formula 165, Art. 1971, and sub- 
stituting, 



D = 



P J 125,000 

ro e 



?854 X 2,800 X .985 



• 7854 4-i£») 

7.596, say 7f in. Ans. 

(1050) Applying formula 163, Art. 1969, and sub- 
stituting, 

P = . 7854/7'/ (l -J^) = .7854 X 6 2 X 1,200 X .99 = 

33,590 lb. Ans. 

(1051) Applying formula 172, Art. 1977, and sub- 
stituting, 

D x = 1.128 V^= 1.128 j/ ~ = 1.189 ft., say 14^ in. Ans. 

(1052) Applying formula 169, Art. 1974, and sub- 
stituting, 

W % = .7854/?,'/ - W x = - 7854 x 162 X 3 > 000 - 5 ^ 350 = 
597,837 lb. Ans. 



HYDRAULIC MACHINERY. 325 

(1053) (a) Applying formula 167, Art. 1973, and 
substituting, 

W x +TV t _ 5,350 + 597,837 _ 



.7854 X 16 2 X .9945 



1 V 100,/ 
3,016.6 lb. per sq. in. Ans. 
(b) Applying formula 168, Art. 1973, and substi- 
tuting, 

_ W 1 + W 2 _ 5,350 + 597,837 

.7854^(l +I Q - - 7854Xl6,><1 - 0055 ~ 
2,983.56 1b. per sq. in. Ans. 

(1054) Applying formula 165, Art. 1971, and sub- 
stituting, 

n - / P+W _ J 15,000 + 3,240 

"/ .7854/ (l~) - 7854 X 50 ° X •"* 5 ~ 

6.841, say 7 in. Ans. 

(1055) Applying formula 164, Art. 1970, and sub- 
stituting, 

P +W_ 52,000 

or 1,056 lb. per sq. in., nearly. Ans. 

(1056) See Art. 1958. 

(1057) See Art. 1963. 

(1058) See Art. 1949, 



/_ 7854 D* L ^\~.7854X8 2 X,98- 1 ' 055 - 6 ' 



Water Supply and Distribution. 

(QUESTIONS 1059-1118.) 



(1059) Wholesomeness, abundance, good pressure, and 
cheapness. Art. 1979. 

(1060) (a) Those of the typhoid and typhoid malarial 
classes, (b) During the visitation of epidemics, such as 
cholera and yellow fever. Art. 1980. 

(1001) No; it is only one of a group of data from which 
intelligent inferences may be drawn. Art. 1981. 

(1062) (<?) It should be clear, colorless, tasteless, odor- 
less, and sufficiently soft, (b) Contamination by even an 
infinitesimal portion of the sewage emanating from an in- 
fected district. Art. 1982. 

(1083) Sewage. Art. 1983. 

(1064) The positive knowledge that it has or has not 
been so contaminated. All other knowledge, however de- 
rived, whether by chemical analysis or otherwise, merely 
arouses or allays suspicion, but gives us no certainty, one 
way or the other, unless in extreme cases of pollution. Art. 
1983. 

(1065) (a) Sewage contamination. (b) Because it 
is an important constituent of human urine, which, in turn, 
is a principal element of sewage, (c) 3 to 10 parts by 
weight per million. Art. 1984. 

(1066) Wanklyn's albuminoid ammonia process, and 
the moist-combustion process. Arts. 1986 and 1987. 

(1067) {a) When it yields also more than 0.05 part 
of albuminoid ammonia per million. Art. 1986. (/>) Un- 
der 0.10 part per million. Art. 1986. 

For notice of copyright, see page immediately following the title page. 



328 WATER SUPPLY AND DISTRIBUTION. 

(1068) (a) By the oxidation of its impurities, (b) The 
nitrites contained in sewage, upon being oxidized, are con- 
verted into nitrates. Nitrites arouse suspicion by indica- 
ting possible previous sewage contamination, and when 
found in large quantities in a water, unless their presence 
can be satisfactorily accounted for, often lead to its rejec- 
tion as a potable supply, but authorities differ regarding the 
interpretation of the presence of this constituent. Art. 
1989. 

(1069) (a) By the soap method, using a standard sapo- 
naceous solution, and noting the number of parts of water 
necessary to produce a " lather " with a given quantity of the 
solution, (b) 50 for soft, 150 for hard. Art. 1992. 

(1070) (a) 100 gallons, {b) By means of tanks in the 
houses or establishments supplied, or by introducing a 
short length of very small pipe in the service pipes, (c) A 
separate high service, using different mains, (d) Fire. 
Arts. 1995 to 1997. 

(1071) {a) By the general use of meters. (b) The 
Venturi meter. Art. 1999. 

(1072) (a) The ocean, and all large exposed water 
surfaces, but chiefly the first named, (b) Evaporation and 
rainfall. Arts. 2000, etc. 

(1073) Part runs off the surface of the ground to the 
most accessible stream or watercourse, and part soaks into 
the ground. Arts. 2001 and 2002. 

(1074) As a great storage reservoir. Art. 2003. 

(1075) Surface and ground water. Arts. 2004 to 
2006. 

(1076) All water taken from lakes, ponds, streams, 
and rivers. Art. 2005. 

(1077) All water taken from beneath the surface of the 
earth by wells, deep or shallow, and springs, when taken 
near the point at which they issue from the ground. Art. 
2006. 



WATER SUPPLY AND DISTRIBUTION. 329 

(1078) Surface water is generally softer, and is more 
subject to contamination than deep-well water. Ground 
water is apt to be harder, from impregnation with earthy 
salts, and in deep wells the temperature is frequently high. 
Shallow-well water is apt to reunite nearly all the bad 
qualities of both sources except the high temperature. 
Springs, on the contrary, frequently furnish the very best 
class of water, but the supply drawn from this source is 
usually limited. Arts. 2005 and 2006. 

(1079) By pumping directly from some large river into 
a distributing reservoir, or, more rarely, by gravity from the 
same source; by damming some smaller stream, thus form- 
ing a reservoir from which water can be taken by pumping, 
if below the level of the town to be supplied, or by gravity, 
if it has been possible to erect the dam at a sufficiently 
high elevation. These two methods relate to surface 
water. The supply might also be drawn from the ground, 
in which case the means employed would be driven wells 
(discarding shallow wells as not suitable for a large public 
supply), galleries, or the collecting of springs in suitable 
basins. Art. 2007. 

(1080) 1st. By gauging the flow of the stream. 2d. 
By ascertaining the area of its water-shed and the yearly 
precipitation or rainfall. Arts. 201 1 and 2012. 

(1081) From 40$ to 50$. One million gallons per 
square mile of drainage area per 24 hours is a convenient 
approximate rule for the available supply from a given 
stream in these States. Arts. 2012 and 2013. 

(1082) To equalize the discharge of the stream, con- 
verting a yearly average into a daily one. Art. 2016. 

(1083) Forty per cent, of 33 is 13.2 inches available 
yearly rainfall. One inch over a square mile gives 17, 378, 743 
U. S. gallons; therefore, 13.2 inches give 229,399,407 gallons 
per square mile. There are &§■ square miles; consequently, 
in round numbers, the yearly yield of the stream is 1,950 
millions. 



330 WATER SUPPLY AND DISTRIBUTION. 



The yearly consumption is 3 X 365 = 1,095 million gallons. 
Then, by the formula (see note, Art. 2017), ^ 1,Q9 ^" = 

JL, JDO 

614.88 million gallons. This is a little over 200 days' supply. 

(108-4) Upon the existence of a water-bearing stratum, 
lying between two impermeable ones. When the upper im- 
permeable stratum is pierced, and a water-tight tube in- 




Impermeable Formation 

Fig. 77. 

serted in the bore, the water rises to a greater or lesser 
height in the tube, as shown in Fig. 77. See Art. 2020. 

(1085) The difficulty consists in the necessity of lift- 
ing the water from each separate well to a common re- 
cipient, from which it can be pumped. Art. 2022. 

(1086) Springs constitute a kind of natural flowing or 
artesian wells. As already stated, in answer to question 
1078, they are apt to be of low temperature, but in this 
country have been seldom utilized for public supplies, owing 
to their rarity, and also, perhaps, because attention has not 
been sufficiently directed to this source of supply. The best 
examples of water supplies on a large scale drawn from 
springs are to be found in the water-works system of 
Paris, France. In developing springs, the greatest care is 
necessary to avoid diverting them to other outlets. Art. 
2023. 

(1087) By exposure to the atmosphere, by which a very 
slow oxidation takes place of the organic impurities. The 
action is very slight, and by no means adequate to purify 



WATER SUPPLY AND DISTRIBUTION. 331 

a water containing any considerable amount of such im- 
purities. Art. 2025. 

(1088) (a) Filtration, either natural or artificial, (b) 
By nitrification, (c) By infiltration through soil, and the 
agency of a bacillus. See Art. 2026. 

( 1 089) (a) See Art. 2G28 and Fig. G49. (b) The char- 
acteristic action of the filter-bed is the formation of a film, 
deposited by the water itself, and constituting the habitat 
of the bacillus referred to in answer to question 1088. (c) 
When the water treated is distinctly turbid. One day's 
supply, (d) When the mean January temperature is below 
freezing, (e) One and four-tenth acres always in use, 
besides the area laid off for cleaning. Arts. 2028 to 
2033. 

(1090) (<?) The prevention of many water-borne dis- 
eases, notably cholera and typhoid fever, (b) Those of all 
large rivers and all suspected waters when the death-rate 
from typhoid fever of the community using them becomes 
excessive. Art. 2035. 

(1091) The substitution of a coagulant, ordinarily 
common alum, for the natural sedimentary film already 
mentioned in the answer to question 1089. (b) Art. 2036. 

(1092) By damming a stream. Art. 2040. 

(1093) To prevent vegetable growth, chiefly that of 
the algae. Art. 2041. 

(1094) Sufficient capacity and elevation. Art. 2042. 

(1095) The use of a second stationary barometer and 
thermometer at some point of which the elevation is known, 
or which may be used as a convenient datum, which should 
be observed at stated intervals. When barometric and ther- 
mometric readings are taken by the explorer, he should note 
the time at which they are taken, so as to compare his 



332 WATER SUPPLY AND DISTRIBUTION. 

readings with those of the observer at the stationary in- 
struments. Art. 2043. 

(1096) (a) By a contour map, made from a careful 
cross-sectioning of the entire probable site of the dam. 
(b) The establishing of permanent monuments. Arts. 
2043 and 2044. 



(1097) When a solid rock foundation for bottom and 
sides of dam can be secured. Art. 2045. 

(1098) (a) A substantial center wall, carried down to 
a suitable depth, (b) To furnish an impermeable cut-off 
against any percolation which might penetrate the dam, 
and also a means of connecting all the appliances, pipes, 
culverts, etc., destined to establish communication between 
the inside and outside of the reservoir, without danger of 
producing leaks. Art. 2046. 



r 






-*- 



4 



■*— 



■ — 10-^ 



23 ■ 12 



Fig. 



D = i 



47 2 X 18 



16 



(1099) See Art. 2046 and Fig. 78. 

(1100) (a) Two and a half horizontal 
to one vertical, (b) Three and a half hori- 
zontal to one vertical, (c) Equal to the 
height of the spillway above the bottom of 
the stream across which the dam is built. 
Art. 2047. 

(HOI) It should be thoroughly cleared 
and grubbed, and all roots and stumps re- 
moved. Art. 2047. 

(1102) Applying formula 173, Art. 
2048, we have the length L = 20 4/I8 = 
85 feet, nearly. Ans. 

By formula 174, we have the depth 

+ 6* =2.1 ft.-f C. Ans. 



(1 103) (a) For dams not exceeding lo ft., a vertical or 
slightly battering face may be given, with a substantial 



WATER SUPPLY AND DISTRIBUTION. 333 

apron to receive the shock of the falling- water. For higher 
dams, either a curved surface or a series of steps, (b) It 
is more economical than the curved form, which involves 
much stone-cutting, and it is very effectual in destroying 
the momentum of the falling water. Art. 2052. 

(1104) Simplicity, efficiency, and strength. Art. 
2054. 

(1105) To accommodate the sliding sluice-gates, and 
the grooves for the stop-plank and screens. Arts. 2054 
and 2055. 

(1106) Sluice-gates and stop-cocks. Arts. 2054 and 
2055. 

(1107) To maintain a uniform head near the point of 
consumption; to afford a convenient place for the more 
complicated appliances for the control of the water dis- 
tributed to different parts of the town, and to furnish a 
small reserve supply of water in the immediate vicinity of 
the town, to provide against any accidental temporary inter- 
ruption of the pumps. Art. 2057. 

(1108) («) The summit of a conveniently situated 
rounded hill, the ground sloping away in all directions from 
it. (b) In excavation, with concrete bottom and masonry 
side walls. Arts. 2058 and 2059. 

(1109) There are several ways in which this can be 
done; probably the sim- 
plest and best is by clo- 



sing the end of the cham- ffiff* j^ ifj^ll} 

ber C, and putting in jL, 

sluice-gates at different 

levels, as shown in Fig. 79. JI 

(lllO) That they FlG - < 9 - 

have a sufficient cross-section to insure against crushing 
under the weight of the cover. Art. 2061. 




334 WATER SUPPLY AND DISTRIBUTION. 



(1111) That the horizontal thrust of the water is always 
equal to 31.25 H 2 , If being the head of 
water sustained by the dam ; and that 
the moment of the same, around the 
outer toe of the dam, is 10.42 H 3 . Art. 
2063. 

(1112) {a) By the moment of the 
thrust, as given in previous answers. 
{b) By its moment of resistance around 
its outer toe. Arts. 2064 and 2065. 

(1113) The section of wall being 
as shown in Fig. 80, and the density 

being 125, the resisting moment about the outer toe B is 




180 X 16 = 

30 X 13 2 , , 

X :r X 13 

Z 6 



2880 
13 2 X 10 = 1690 



4570 
125 

22850 
9140 
4570 

571250 
The overturning moment about the same point is 
10.42 X 30 3 =281,340. 
57,125 



Then. 



2.03 



factor of safety. 



28,134 

Inserting data in formula 1 82, Art. 2068, 
19 =£4/9,00 C +108 — 3. 
22 2 =1(900(7+ 108). 
484 = 225 C + .27. 

C = fir = 2-03 = factor of safety, as above. 



WATER SUPPLY AND DISTRIBUTION. 335 



(1114) Resisting moment of plumb wall, neglecting 
density, is |~6_ 

27 X 18 X 18 _ 

2 -*,*<*• 

Resisting moment of trapezoidal wall, as 
per Fig. 81. is 

27 X 6 X 15 = 2,430 



Then, 



e^ x | xl , 



Trapezoidal 



1,296 

3,726 
3,726 



-6 



^ 



18 1 

Fig. 81. 



0.85. 



Plumb 4,374 

The resistance to sliding of plumb wall, neglecting co- 
efficient of friction, is 

27 X 18. 

That of trapezoidal wall is 

w (?±li) =*»*«. 

Trapezoidal 



Hence, 



12 

r8 = 0.67. 



Plumb 

(1115) Crushing of the material. Art. 2071, 

(1116) Taking moments about the toe B, the resultant 
of the weight of the whole mass above A B when the reser- 
voir is empty is obtained thus: 

300 X 56.67= 17001. 
3675 X 46.67 = 171512.25 
337.5 X 73 = 24637.50 



4312.5 


213150.75 


4312,5 


144 

17250.0 


172500 
40650.7 


49.43 


172500 


38812.5 




43125 


1838.25 




621000.0 


1725.00 




113.250 




121). 37, 


5 



336 WATER SUPPLY AND DISTRIBUTION. 



Then 79 -49.43 




29.57; and we get the diagram, Fig. 82. 
Proceeding as in general 
example (see Art. 2075), 
we get, max. stress in seg- 
ment 



A = 



621,000 X 49.43 



79 X 29.57 
13,140 lb.; 
max. stress in segment 
B _ 621,000 X 49 _ 

~~ 79 X 30 ~~ 
12,839 lb. 

(1117) The resisting 

344531.25 V ' t 1 S 

fig. 8-2. moment of the dam about 

the outer toe B is (utilizing calculations already made in 
previous answer) 

213,150.75 X 144= 30,693,708 ft. -lb. 

The overturning moment, due to the thrust of the water, 
is 

10.42 X 105 3 = 12,062,452.50 ft. -lb. 

Then, factor of safety = 

30,693,708 _ 
12,062,453 - >0 • 

It would not be w T ise, therefore, to reduce the base, 
although it might be safely done as against crushing, 
because the above factor of safety against overturning ig 
not too great. 

(1118) That dams of less than say 125 ft. in height 
must be calculated to resist overturning as well as crushing 
of the material. 



Water Supply and Distribution. 

(QUESTIONS 1119-1143.) 



(1119) The surface of water, when at rest, is level. 
No motion, or flow, is possible unless the surface assumes 
a sloping line. The hydraulic grade-line is the grade 
assumed by the surface of flowing water. In an open canal, 
this grade is easily determined; in a pipe-line of uniform 
diameter and character, it is the imaginary line joining the 
upper and lower extremities of the pipe; or, if the pipe con- 
nects two reservoirs, then it is the line joining the two water 
surfaces. If the line is composed of pipes of different diam- 
eters, or if there are branches leading to or from the pipe- 
line, then the hydraulic grade becomes a broken line, and it 
is located by determining the piezometric height at each 
point where any change occurs in the character of the line. 
Art. 2085. 

(11 20) A reducer, in order to diminish the resistance to 
entry, and consequent diminution of velocity, due to square 
or abrupt angles at the point of entrance, or at the change 
of diameters of the connecting pipes. 

(1121) (a) A vertical tube, usually an imaginary one, 
open at the top, and supposed to be inserted in a pipe through 
which water flows, in such a way that the water in the main 
pipe may freely rise and fall in it. (&) The height to which 
the flowing water does or may rise in a piezometric tube in- 
serted, or supposed to be inserted, at the given point, (c) A 
pressure-gauge ; the recorded pressure can be converted into 
the piezometric height, expressed in feet, by multiplying the 
pounds per square inch by 2.304. Arts. 2085 to 2087. 

(1122) To assure ourselves that the pipe-line does not 
rise above it at any point. Art. 2087. 

For notice of copyright, see page immediately following the title page. 



338 WATER SUPPLY AND DISTRIBUTION. 

(1123) Direct experiment. Arts. 2088 and 2089. 

( 1 1 24) One of w hich the length is at least one thousand 
diameters. Art. 2092. 

(1 125) See formula 186, Art. 2092. 

(1126) See formula 188, Art. 2092. 

(1127) (a) See formula 193, Art. 2094. (&) See 
formula 196, Art. 2094. 

(1128) 13.5 X 1.4= 18.90 cu. ft. per sec. Art. 2095. 



(1 1 29) (a) Q = yTx 1 & = 1 cu. ft. per sec. 

(b) Because it enables us to deduce, readily, the corre- 
sponding elements of any other pipe, by means of the relation 
given in formula 205, Art. 2097, which is sometimes a 
convenience. 

(1130) (a) Here we have: Z>=1; #=1.3725; L = 

1,372.5, and, by calculation, Q = 1. Also, we have : D' = 
1.5; H' = 4.6, L' = 2,300. Q is required. Inserting these 
data in formula 206, Art. 2097, 

l 2 X 1,372.5 XlTx 4.6 



Q % X 2,300 X l 5 X 1.3725 



w u a f 2x 7.6 . 

which reduces to — -y^ — =1; 

whence, Q' = 3.90 cu. ft. per second. Ans. 



/ax n i/7.6 X 4.6 X 1,000 . , . 

(b) Q = y ■ — = 3.90 cu. ft. per second. Ans. 

2, o00 

(1131) From formula 191, Art. 2094, 

3A2 -y 17,600 • 

Hence, 11.70 



17.6 

and, D = 1/2.744. 

£>= 1.224 ft 
Or say 16 inches. Ans. 



WATER SUPPLY AND DISTRIBUTION. 339 

(1132) Let us make elevation of the extremity of pipe 
(300 in the question) equal 0. Then, elevation of extremity 
of 6-inch pipe is 20, that of 8-inch pipe 90, and that of 
reservoir 200. We then represent all on a sketch, drawn to 
scale, on cross-section paper. We will adopt the approximate 
method (Art. 21 03), and assume a piezometric height at 
point of embranchment of 6-inch pipe = 30 ft. Then, 
quantity Q discharged at the extremity of 16-inch pipe is 



^ ,/30,000 /4\ 5 „ tf , 

Also, quantity Q' discharged by 6-inch pipe is 
Q' = 4^10 X 05 5 =0.56. 

Total discharge from both = Q + Q' = 7.05. 

We now find the piezometric height, above elevation 30 at 
the embranchment of the 8-inch pipe, necessary to force 
this amount of water through 5,000 ft. of 16-inch pipe, thus : 

*:=-ro54 = 11 - 8 - 



243 

This is the rise per 1,000; hence, total rise = 11.8 X 5 = 
59 ft. 

We now test this value by adding it to the piezometric 
height 30, which gives 59 + 30 = 89 ft. 

As the elevation of the extremity of the 8-inch pipe is 90, 
it is clear that this elevation is too low to produce a flow in the 
8-inch pipe. It is possible, however, that no water can flow 
from this pipe, on account of the height of its point of dis- 
charge. To test this, we carry the rise of 11.8 ft. per 1,000 
through to the reservoir, which is distant 12,000 ft. from the 
point of embranchment of 6-inch pipe. This gives 11.8 X 
12 = 141.6 ft., which, added to the assumed piezometric 
height, 30 ft., gives 171.6 ft. But the elevation of the 
reservoir is 200 ft., so the assumed height of 30 ft. is not 
high enough, and if raised sufficiently to reach the reservoir 
there would be a discharge through the 8-inch pipe. 

Q. G. IV.— 21 



340 WATER SUPPLY AND DISTRIBUTION. 

We, therefore, try another piezometric height, 33 ft., which 
gives 

Q = VV X (i) b = G.77 cu. ft. 

Also, the discharge through 1,000 ft. of 6-inch pipe, the 
fall being 33 - 20 = 13 ft. per 1,000, is 

Q = 4/13 X 0.5 5 = 0.64 cu. ft 

Then, g + Q' = 7.41 cu. ft. 

For the 5,000 ft. of 16-inch pipe, the necessary rise per 
1,000 will be 

7 41 2 

243 

The total rise is, therefore, 13.03 X 5 = 65.15 ft. 

This being added to the assumed height of 33 ft. gives 
98.15 ft. for the piezometric height at the junction of the 
8-inch pipe with the 16-inch. 

The 8-inch pipe is 1,200 ft. long, and the elevation of its 
extremity is 90 ft. Hence, the rise per 1,000, /i, is 

h= 98.15- 90 =6g 

1. z 

Then the quantity Q" which it discharges is 



Q" = 4/6.8 X HY = .95 cu. ft. 

The total discharge is, therefore, 6.77 + 0.64 + 0.95 = 
8.36 cu. ft. 

We now want the rise per 1,000 to discharge this through 
the remaining 7,000 ft. of 16-inch pipe which intervenes 
between the 8-inch pipe and the reservoir. 

We have h = -f^- = 16.59 ft. 

243 

On 7,000 ft. this amounts to 16.59 X 7 = 116.13 ft. This, 
added to 98.15, gives 214.28 ft. As the elevation of the 
reservoir is only 200 ft., we have taken our first assumed 
height too high. Our first assumption, 30 ? was too low, the 



WATER SUPPLY AND DISTRIBUTION. 341 

present one, 33, is too high, so the true value evidently lies 
within narrow limits. 

An inspection of the above results shows that an addition 
of 3 ft. to the first assumed piezometric height made a 
difference of 214.3 — 171.6 = 42.7 ft. in the required eleva- 
tion at the reservoir. 

The difference for 1 ft. is, therefore, 42.7 -4- 3 = 14.2 ft. ; 
consequently, if we reduce the assumed piezometric height 
from 33 ft. to 32 ft., the required height at the reservoir 
will be reduced to about 200 ft., which is the value it must 
have to agree with the data. 

Using 32 ft. as the first assumed piezometric height, and 
calculating as before, we have 



Q =f¥X (*) 6 = 6.7 



0'=Vl2 X .o 5 = .6 
Q + Q ="^¥cu. ft. 

Fall per 1,000 in 5,000 ft. of 16-inch pipe, 

243 
12.64 X 5 + 32= 95.2. 



G+ Q + Q" = 6.7 + .6 + .76 = 8.06 cu. ft. 
Fall per 1,000 in 7,000 ft. of 16-inch pipe, 
i ; 8.06 2 1K ; 1f . 

243 

15.41 X 7 = 107.87 ft., 

which, added to 95.2, gives 203.07 ft., as against 200. This, 
in practice, would be a satisfactory result, and the calcula- 
tion would be carried no further, because, at best, all these 
calculations are but approximations, indicating the probable 
£ruth. 



342 WATER SUPPLY AND DISTRIBUTION. 

(1133) Applying formula 209, Art. 2111, we have 
4,000 _ 1,000 ,700 ^00 1,500 

12 ^°° . = 243,000 + 22,400 + 800 + 1,536,000. 

128,000 - 

1,802,200 
Log- D h = 2.851258. 
Log D =1.770251; D = 0.589 f t. = 7 + inches. Ans. 

(1134) See Art. 21G6. By making a sketch on sec- 
tion paper, it will be evident that the two upper reservoirs 
feed into the lowest one; also, that the piezometric height 
at the junction of the 12-inch pipe with the 18-inch will be 
about 380. Calling the elevation of the lowest reservoir 0, 
gives elevation highest reservoir 250, elevation of interme- 
diate reservoir 150, and piezometric height 130. 

Then, quantity received in lowest reservoir is 



Q = 4/37.14 X 7.6 = 16.80 cu. ft. 
Quantity delivered by highest reservoir, 

Q! = 4/34.29 X 776 = 16.10 cu. ft. 
Quantity delivered by intermediate reservoir, 



Q" = 4/4^44 = 2,11 cu. ft. 

Q ,J r Q" = 18.21 cu. ft., total received by lowest reservoir. 
Hence, the assumed piezometric height of 130 is a little too 
low. We try 135. 



Then, Q = 4/38.57 X 7.6 = 17.12 cu. ft. 

Q — 4/32.86 X 7.6 = 15.80 

(7 = 4/3733 = 1.82 

■ 17.62 cu. ft. 

This is a very satisfactory check, and we carry the calcu- 
lation no further. 



WATER SUPPLY AND DISTRIBUTION. 313 



(1135) See Art. 2113. Area of pipe, 3.1410 sq. ft. 
elocity necessary 
per second. Then, 



23 

Velocity necessary to produce given flow, 7—7— = 7.32 ft. 

0. 1410 



Frictional height = ( 23 ) 2 x 30 = .49G 

s 32 X 1,000 

3 X (7 32) 2 
Velocity and entrance heads =± — — ' J rt = 1.249 
J 4 X 32.16 



1.745 ft. 

This would bring the top of the pipe so near the surface 
of the water that it could be scarcely considered a good 
arrangement, because vortexes might be produced. It 
would be better to use a somewhat smaller pipe, <-ay 20 or 22 
inches, with a deeper submersion. 

(1136) Sudstituting data in formula 2 IO, Art. 2117, 

7 H 

350 "IT 

77=440. Subtracting the height of reservoir above 
pump, 440 — 360 = 80 ft. The distance being 3,000 ft., we 
have 

h = *£ = 27, nearly. 

Then, D = 4/ff =1.127 ft. , or say 14 inches. 

(1137) Daily supply, 3>0 ^ 00 = 401,070 cu, ft. per 

24 hours. If consumed in 10 hours, this gives 40,107 cu. ft. 
per hour, or 11.14 per sec, for which discharge the mains 
should be calculated. See Art. 2121. 

(1 138) ^'^ = 11.36 = distance in miles. 

Head, 250 ft. Then, thickness of pipe from formula 214, 
Art. 2128, is 

T= 0.00006 X 250 x 10 + 0.0133 X 10 + 0.296 = 0.579 inch. 

Then, from formula 213, Art. 2127, 

P= 25 X 11.36 X 10.579 X 0.579 = 1,740 long tons. Ans. 



344 WATER SUPPLY AND DISTRIBUTION. 

(1139) Air-cocks at the summits and blow-offs at the 
valleys, or lowest points. Arts. 2133 and 2134. 

(1140) To start with all air-cocks and blow-offs open 
and fill very gradually, closing blow-offs and air-cocks suc- 
cessively, as described in Art. 2137. 

(1141) A distributing reservoir, containing a week's 
supply, situated at such an elevation as to give an efficient 
pressure throughout the town, which distributing reservoir 
is fed from a storage reservoir by gravity. Art. 2138. 

(1142) 42 X 1.05 = 44.10 inches. 42 X 1.10 = 46.20 
inches. 45 inches would in ordinary cases be the proper 
diameter. Art. 2141. 

(1143) Applying the rules of Arts. 780 to 785, we 
find the water section 5 and the wetted perimeter WP as 
follows : 

The water section is found by dividing the section of the 
aqueduct into three parts: a semicircle whose diameter is 
3.71 X 2 = 7.42 feet; a segment whose radius is 8 feet; rise, 
.75 foot, and chord, 6.75 feet; and the trapezoid included 
between the semicircle and the segment. 

The water section included in the semicircle is equal to 
the area of the semicircle minus the area of the segment 
whose rise is 1.12 feet. The area of the semicircle is 

7.42 2 X .7854 ^ rl ■ 
— 21.62 square feet. 

In order to find the area of the segment, we first find the 
angle included by its arc. The cosine of one-half of the arc is 

371 - 1 - li = . 69811; 



3.71 

from the table of sines and cosines the corresponding angle 
is found to be 45° 43V ; therefore, the angle included by the 
arc is 2x45° 43V = 91° 27' = 91.45°. The area of the 
included sector is 

43.24 X 91.45 



360 



10.98 square feet. 



WATER SUPPLY AND DISTRIBUTION. 34.5 
The length of chord of the arc is 



2 4/3. 71 2 - 2.56 2 = 5. 312 feet, 

and the area of the triangle included between this chord and 
the radii to its extremities is 

5.312X2.59 n QQ , . 
■ = 6.88 square feet. 

2 

The area of the segment is, therefore, 10.98 — 6.88 = 4.10 

square feet, and the area of the water section included in 

the semicircle is 21.62 — 4.10 = 17.52 square feet. 

To find the area of the lower segment, we first find the 

angle included by its arc. The sine of one-half of the angle 

3 375 > 
is — '— — = .42188. The corresponding angle is 24° 57', 
8 

nearly ; therefore, the angle included by the arc of the seg- 
ment is 49° 54' = 49.9°. The area of a circle whose radius 
is 8 feet is 16 2 X .7854 = 201.06 feet. The area of the 
sector is, therefore, 

201.06X49.9 

— = 27.869 square feet; 

the area of the triangle included between the chord and the 
radii to its extremity is 

6.75 X 7.25 £) . Aa . , ^ 
= 24.469 square feet, 

2 

leaving 27.869 — 24.469 = 3.40 square feet as the area of the 
segment. 

The area of the trapezoid included between the semicircle 
and the lower segment is 

7.42+6.75 . no _. 

l - — X4 = 28.34 square feet. 

2 

The total area of the water surface is, therefore, 

5=17.52 + 3.40 4-28.34= 49. 2G square feet. 



346 WATER SUPPLY AND DISTRIBUTION. 

The perimeter of the semicircle is 

3.1416 X 7.42 

2 
contact with the water is 



11.655 feet, and the length of the arc not in 
water is 
23.31 X 91.45 



360 



5.921 feet. 



This gives us 11.655 — 5.921 = 5.734 feet as the wetted per- 
imeter included in the semicircle. The circumference of a 
circle whose diameter is 16 feet is 16 X 3.1416 = 50.265 feet; 
therefore, the wetted perimeter included by the lower seg- 
ment is 

50.265 X 49.9 = 6967feet 

The wetted perimeter included by the trapezoid is 2 X 4 = 
8 feet, very nearly. 

The total wetted perimeter is 

WP= 5.921 + 6.967 + 8 = 20.888 feet 

The mean hydraulic radius is 

„ 5 49.26 op 

Applying formula 215, Art. 2143, we have 



TT T>A7 100,000/ nnnA / 21 a^r 

Ans. 



IRRIGATION 

(QUESTIONS 1144-1217.) 



(1 144) It is still in its infancy, but constantly growing 
in importance and in scientific methods. (Art. 2147.) 

(1145) Because plants require that the substances 
necessary for their growth should be presented to them in 
solution in water. They drink, and do not eat, their nour- 
ishment. (Art. 2148.) 

(1146) By rainfall. (Art. 2149.) 

(1147) East of the 97th meridian. (Art. 2149.) 

(1 148) A sufficient but not excessive quantity of water, 
supplied with regularity at certain stages of plant growth, 
depending upon the nature of the crop and its time of 
planting and harvesting. (Arts. 2149 and 21 50.) 

(1149) A proper public water supply requires to be so 
regulated and controlled that a certain daily quantity 
throughout the year shall be furnished. The answer to the 
previous question shows that for irrigation purposes the 
supply is intermittent, and is needed at certain seasons only. 
(Art. 2153.) 

(1150) Artificial irrigation, when properly conducted, 
is superior to rainfall in that it can be supplied or withheld 
at will, according to the needs of the crops cultivated. 
(Arts. 21 50 and 2151.) 

(1151) From 40,000 to 100,000 cubic feet per acre per 
annum, or say sufficient to cover the area irrigated to a 
depth of from 1 to 2£ feet. (Art. 2152.) 

For notice of copyright, see page immediately following the title page. 



348 IRRIGATION. 

(1 152) Yes; because such favored regions are still sub- 
ject to periods of drought at seasons when crops most need 
water. (Art. 2149.) 

(1153) See Art. 2152. 

(1154) Provided the water does not contain distinctly 
injurious substances, such as those favoring the production 
of alkali, quality is a matter of small moment. Indeed, 
water charged with organic matter, and even sewage, is 
thereby rendered more beneficial to plant growth than if 
perfectly pure. (Art. 2153.) 

(1 155) (a) Upwards of 40,000,000 acres, (b) $450,000,- 
000. (Art. 2154.) 

(1 156) A very important part. In many cases irriga- 
tion without underdrainage, either natural or artificial, 
would be an injury instead of a benefit. (Art. 2155.) 

(1157) The presence of alkali is a serious detriment to 
the growth of crops, covering the surface of the ground with 
a hard efflorescent crust composed chiefly of salt, sal soda, 
and Glauber's salts, of which the sal soda, known as "black 
alkali," is the most pernicious. (Art. 2156.) 

(1158) Thorough underdraining to prevent super- 
saturation of the soil, mulching, and the use of gypsum as a 
top dressing. Sometimes it can be washed off by flooding. 
(Arts. 2156 and 2157.) 

(1159) No. It should be supplemented by all the 
other resources known to the agriculturist. Irrigation merely 
substitutes rainfall, leaving all the other requirements as 
before. (Art. 2158.) 

(1160) (a) Surface and ground water, {b) Surface 
water, as used in irrigation, consists in the run-off of streams ; 
ground water is all water derived from wells, deep or 
shallow. (Art. 21 60.) 

(1161) The quantity of water required, if the area to 
be irrigated is given ; or the area that can be irrigated, if the 
amount of water disposable is given. (Art. 2160.) 






IRRIGATION. 349 

(1 162) 23 X 640 X 175,000 = 2,576,000,000 cu. ft. Ans. 

(1163) {a) The run-off of a given area is the unabsorbed 
and unevaporated volume of rain falling upon such area, 
which finds its way to its brooks, streams, and rivers, (b) 
From 33^ to 500. (Art. 21 60.) 

(1164) Diameter of funnel, W 5 - inches; diameter of 
cylinder, J^. Then the actual depth of rainfall is to depth 
of water in cylinder (3.17 inches) as the square of diameter 
of cylinder is to square of that of the funnel. Representing 
the desired depth of rainfall by x, the above proportion will 
be written: 

*:3.17 = (W-) 2 :(-W) a . 
Whence, ;r = 0.397. 

That is, the rainfall was 0.397 inch for the H\ hours, or 
0.0541 inch per hour. Ans. 

(1165) By means of a weir. (Art. 2163.) 

(1166) Applying formula 217, Art. 2163, we have 
Q — 3^ x 4.23 X 1.73* = 32.08 cu. ft. per sec. Ans. 

(1 167) All observations must be made with great care, 
and they should be repeated until it is certain they are 
as near correct as the conditions will allow. (Art. 2164.) 

(1168) These difficulties consist in the fact that the 
evaporation from the surface of a large body of still water 
is different from that of the supplying stream. (Art. 
2165.) 

(1169) From 3 to 5 feet per annum. (Art. 2165.) 

(1170) Generally, facilities for discharging large vol- 
umes of water during short periods of time are more neces- 
sary for the irrigation reservoir than for water supply. 
(Art. 2166.) 

(1171) This sketch should be made according to the 
general features shown in Fig. 701, Art. 2167, carrying 
the cribs well down to the good foundation. 



350 IRRIGATION. 



(1172) 7—r^r = 0.00113(3 = tangent of the angle of the 
5,280 

slope. The nearest number given in the table is 0.0011G, 

corresponding to an angle of 4 minutes, which would be 

sufficiently close in such preliminary work. 

(1 173) The character of the material through which it 
runs, regarding its greater or less liability to cave in and 
wash, and the consequent admissible velocity. (Art. 
2171.) 

(1174) Gravity. (Art. 2172.) 

(1175) See Art. 2173. 

(1176) {a) Since the slope of the sides is 1^- horizontal 
to 1 vertical, the top width of the water section is 8 + 
2 X 5 X 1| = 23 feet, and the length of each of the sloping 
sides is |/o 2 + 7V = 9.01 feet; the length of the wetted 
perimeter is, therefore, 8 -f 2 X 9.01 = 20.02 feet. Ans. 

23 I 8 

(b) The area of the water cross-section is J~ X 5 = 77^ 

square feel Ans. 

(c) The hydraulic radius is 77.5 -f- 20.02 = 2,98 ft. Ans. 

7 

(d) The slope is s = „ . =.0013; therefore, by applying 



_ Vl00,000 X 2.9S 2 X .0013 



formula 21 9 9 Art. 2177, we have the velocity of flow 

9X2.08 + 35 4.32 ft. per sec. 

The discharge is, therefore, Q — a v — 77.5 X 4.32 = 334.8 
cu. ft. per sec. Ans. 

(1177) Let x— depth of water. Then, %x= width of 
flume. The wetted perimeter will then be 4.r, and the 
area of the water cross-section 2 x 2 . The hydraulic radius 

r = \^- = 0. 5 x. The slope is s = -^tt = • 0017. The dis- 
Ax r 5,280 



IRRIGATION. 351 

charge is to be 100 cu. ft. per second. The mean velocity 

must then be 

_ 100_ 50 

V ~ 2x~* ~~ .?" 
Inserting the data in formula 226, Art. 2196, 



50 _ ./ lOO^OO X 0.25 x 2 x 0.0017 
x*~* 3.3* + 0.46 

Squaring both sides, 

2,500 _ 100,000 X 0.25 x 2 x 0.0017 
* 4 ~~ 3.3* +0.40 

1 __ 0.017* 2 
* 4_ 3.3* + 0.46' 
0.017 * 6 - 3.3* = 0.46. 
x 6 - 194* = 27. 
Assuming a value of x = 3, the first number of the equation 
becomes 

3 6 - 194 X 3 = 729 - 582 = 147, 

which is too large. Trying 2.9, we have for the value of 
the first number, 

2.9 6 - 194 X 2.9 = 594.8 - 562.6 = 32.2 ; 

and 2.8 gives us 

2.8 6 - 194 X 2.8 = 481.9 - 543.2 = - 61.3. 

We therefore see that the true value is slightly less than 
2.9. We will make the depth 3 feet, the width of the flume 
6 feet, and check our work by computing the quantity that 
would be discharged by our flume. Substituting in formula 
226, Art. 2196, we have, since the wetted perimeter is 
2 X 3 + 6 = 12 ft., the water cross-section 18 sq. ft., and the 
hydraulic radius 18 -=- 12 = 1.5. 



v = j/- 



100,000 X 1.5" X .0017 __„^ A 

= 0.07 it. per second. 



6.6 X 1.5 + .46 
The discharge, therefore, will be 

18 X 6.07 = 109.26 cu. ft. per sec. 
which satisfies the required conditions. 



352 IRRIGATION. 

(1178) The total weight of the water in the flume is 
6 X 3 X 15 X 62.5 = 16,875 pounds. Weight of timber, as- 
sumed at 10 per cent, of the water, 1,687 pounds. Total 
weight to be carried by the stringers, 16,875 + 1,687 = 
18,562 pounds, half of which, or 9,281 pounds, goes to each 
stringer. 

Since the stringers are to have a square cross-section, 

d % 
b = d and formula 222, Art. 2189, becomes W— %-j S. 

From the table in Art. 2193 we find the allowable unit 
stress 6* for white oak with a steady load to be 1,350 pounds. 
Substituting in formula 222, we have 

»-*»l = 4X 15^12 Xl ' 35 °> 

from which d 3 = 928.1 and ^=9.61, from which we see 
that the section of the stringers should be 10" X 10". Ans. 

(1 179) The total uniformly distributed load is 1,500 X 
36 = 54,000 pounds. We will assume the depth d of the tie- 
beam to be 14 inches; then, the allowable unit stress for 
spruce, from the table in Art. 2193, being 1,250 pounds, 
we have, by substituting in formula 232, Art. 2200, 

54,000X36X12/ 1 



1,250 = J X 



K UJ' 



b X 14 \2 X 10 X 12 ' X 

from which b = 9.32 inches. 

In order to provide for the holes for the queen-rods, we 
will make the tie-beam 12" X 14". 

The total stress in the upper chord member, from for- 
mula 233, Art. 2200, is 

c T 54,000 X 36 X 12 .. OAA A 

s c = ix 10 x 1 2 — = 24 ' 300 p° unds ; 

and, from formula 234, the total stress in each of the 
struts is 



V i + « x ^ 



S a = 54,000 y t + -gV X Jq- 2 = 30,240 pounds.. 
The values obtained for S c and 5 S show that the struts 



IRRIGATION. 353 

carry a heavier stress than the top chord member; we will, 
however, make them all the same size, computing the 
dimensions for the struts only. 

The length of a strut is */l2 2 + 10 8 = 15. 62 ft. = 188 inches, 

nearly. Assuming a depth of 10 inches, a value of -p 2 of 

1,250, and substituting in formula 79, Art. 1171, we have 
the breadth 



3o ; aWi i 12xl882 \ 

30,240 \ 1+ 10" X3,000J 



b — )— ^^ ; = 5.84 inches. 

10 X 1,250 

The struts and top chord member may, therefore, safely 
be made of 6"x 10* timbers. 

Since the queen-rods are to be wrought iron, the safe unit 
stress may be taken as 12,000 pounds per square inch. Sub- 
stituting in formula 235, Art. 2200, the net area is 

. , 30,240 

^ = - x I^ooo-- 84s( i- in - 

From a table of standard screw threads, we find that the 
area at the bottom of the thread of a 1^-inch bolt is .893 
sq. in. ; consequently, a 1^-inch queen-rod will be used. 

(1 180) One panel weight is l,800x 12 = 21,600 pounds. 
By applying formula 236, Art. 2202, to the different 
ties, we have, remembering that the stress in the center tie 
is equal to 1 panel load, 

Stress in c — 21,600 pounds. 

Stress in b—l^X 21,600 = 32,400 pounds. 
Stress in a — 2£ X 21,600 = 54,000 pounds. 



The length of each of the struts is L s = |/8 2 + 12 2 = 14.42 
feet; therefore, the factor -^ in formula 237 becomes 

14.42 , v., • 

— - — = 1.8, very nearly, and the stress m the struts is for 

o 

Strut c\ 21,600 X 1.8 = 38,880 pounds. 
Strut //, 32,400 X 1.8 = 58,320 pounds. 
Strut a\ 54,000 Xl.$ = 97,200 pounds. 



354 IRRIGATION. 

The ratio of the length of the panel to the length of the 
tie is j2 = ±£-= l-i- ; applying the known values to formula 

238, we have for the different top chord members, 

Stress in B' = 2 X 21,600 (3 — 1) 1£ .= 120,600 pounds. 
Stress in ^' = 21,600(3 — ^)1 J = 81,000 pounds. 
Applying formula 239, we find that the stresses in the 

bottom chord members are as follows: 

Stress in C = 21,600 (3 X 2+3— 2—2— i) 1£ = l-±5,800 pounds. 

Stress in^ = 21,600 (3+3— l—i—i)li = 129,600 pounds. 

Stressing = 21,600(3— i) 14 = 81,000 pounds. 

(1181) (a) Since it went down 5 inches in six blows, 
its final set, or refusal, under the last blow was evidently 
less than an inch. Formula 240, Art. 2205, is there- 
fore applicable, and we have 

5 = 3,000 X 8 = 24,000 lb. Ans. 

(b) As the area of the head of the pile is 113 sq. in., and 
its safe resistance to crushing may be taken as 1,000 lb. per 
sq. in., its safe resistance to crushing is 113,000 lb., greatly 
in excess of the value found in the answer to (a) for the safe 
load against further penetration into the ground. 

(1182) Overflow weirs and discharge gates. (Art. 
2208.) 

(1183) In reduced freight, owing to less weight; to 
economy in handling and laying, for the same reason, and 
in the less number of joints. (Arts. 2210 to 2214.) 

(11 84) Low pressures, and cheap supplies of the proper 
kind of lumber. (Art. 2215.) 

(1185) Yes; because loosened fragments of rock are 
always liable to fall and obstruct the passage of the water. 
To remove such fallen fragments, it is necessary to empty 
the tunnel, which may cause embarrassing interruptions of 
service. (Art. 2216.) 

(1186) By the slow saturation of the earth's crust 
through ages of rainfall. (Art. 2217.) 



IRRIGATION. 355 

(1187) A great reservoir, in which the rainfall of 
many centuries has been stored away. (Art. 2217.) 

(1188) Deep and shallow. (Arts. 2218 and 2219.) 

(1189) Into flowing and non-flowing wells. The 
former only being properly called artesian. (Art. 2219.) 

(1190) See Art. 2220. 

(1191) One that develops at least one indicated horse- 
power, by the combustion of two pounds of good coal per 
hour. (Art. 2227.) 

(1 192) When large volumes of water are to be pumped 
with regularity in localities where fuel is dear, and where 
there are facilities for running and maintaining the costly 
and complicated high-duty pumping engine. (Art. 2227.) 

(1193) Irrigation by sprinkling consists in moistening 
the ground by throwing water upon it in the form of spray. 
Its advantages are : that a thorough and gradual saturation 
may be administered to all parts of the field, with no special 
preparation of the ground in the way of leveling and gra- 
ding, and the watering being at all times under perfect con- 
trol. The use of this method is naturally inconvenient and 
expensive when applied to very large tracts requiring large 
volumes of water, owing to the necessity of pipes, hydrants, 
pumps, hose, etc., for its application. (Art. 2232.) 

(1194) The object of this system of irrigation is to 
spread a thin sheet of water as uniformly as possible over 
the land. To accomplish this some preliminary work in the 
way of leveling and grading is almost invariably necessary. 
This being accomplished, water is led by a ditch along the 
upper level of the field to be treated, and by means of tem- 
porary obstacles placed in the ditch, it is caused to slowly 
overflow and spread over the land below it. Or, breaks or 
openings are made at certain intervals in the ditch, through 
which the water runs out and spreads over the lower por- 
tions of the field. The course of the water must be carefully 
watched and directed while it is being laid on. (Art. 
2233.) 

(J. a. IV.— 22 



356 IRRIGATION. 

(1195) («) Land lying on a gentle and regular slope. 
(b) It is the simplest and cheapest method of irrigation, 
but is wasteful of water, and very irregular in its distri- 
bution. (Art. 2233.) 

(1196) By running a number of parallel ditches, one 
below the other, across the slope of the ground, at short 
distances apart, dividing the hillside into zones or belts. 
These belts are successively watered, commencing with the 
upper one, the unabsorbed water passing to the next ditch 
below, and so on. (Art. 2233.) 

(1197) In the check system, the auxiliary ditches just 
described are replaced by ridges, or low dams, rudely 
thrown up. Their object is to submerge the land lying 
between the ridges or checks, and to allow the water to re- 
main upon it until it becomes sufficiently saturated. It is 
then drawn off, and allowed to flood the next zone below. 
The checker-board system is a modification of the above, and 
consists in crossing the checks just described by others, 
running up and down hill, dividing the land into a series of 
compartments, which are flooded successively. Both of 
these methods — particularly the latter — are adapted to very 
level land. (Arts. 2234 and 2235.) 

(1198) {a) In the furrow system the water is led be- 
tween the rows or hills in which the crops are planted, and 
the moistening of the soil takes place by lateral absorption. 
(d) Orchards, and all crops sown in hills or rows. (Art. 
2236.) 

(1199) A careful preparation of the land, good drain- 
age, and cultivation. (Art. 2158.) 

(1200) Economy of water, and combining irrigation 
with drainage, in certain cases. (Art. 2237.) 

(1201) The cubic foot and the second. (Art. 2240.) 

(1202) 1 -T- 38.4 = 0.026 cu. ft. per second. (Art. 
2240.) 



IRRIGATION. 357 

(1203) Securing a proper supply of water, transport- 
ing it to where it is to be used, and distributing it to con- 
sumers in measured volum&s. (Art. 2242.) 

(1204) $10.55 and $84.25. (See Table, Art. 2244.) 

(1205) $15.84; $22.27; $19.00. (See Table, Art. 
2244.) 

(1206) The sewage of a town is generally discharged 
into some neighboring stream or watercourse. This leads 
to the greater or less pollution of the stream, which may be 
the source of supply to some other community, or a feeder 
thereto. Consequently, unless the town sewage can be 
emptied into the sea, or a large bay, or powerful tidal river, 
it becomes necessary, in the interest of public health, to 
purify the sewage before allowing it to empty into any 
watercourse, the contamination of which would be likely to 
be injurious to other communities. (Arts. 2253 and 
2254.) 

(1207) Depriving it of the injurious ingredients which 
it contains, in suspension or in solution, so that the effluent 
may be wholly or partly innoxious, to the extent of allow- 
ing it to be emptied into a source of public water supply 
without seriously polluting it. (Art. 2254.) 

(1208) Chemical precipitation, intermittent nitration, 
and broad irrigation. (Arts. 2254 to 2257.) 

(1209) In the liquid form. (Art. 2253.) 

(1210) By means of sewage irrigation areas, supple- 
mented by filter beds capable themselves of producing cer- 
tain crops. (Arts. 2257 and 2258.) 

(1211) The water supply of the town. (Art. 2259.) 

(1212) As much as 40 to 50 feet. (Art. 226G.) 

(1213) Alfalfa, Indian com sown for forage, garden 
crops, etc. (Art. 2261.) 



358 IRRIGATION. 

(1214) Yes; trouble may be apprehended when the 
mean winter temperature falls for any considerable time 
much below 20° to 25° F. (Art. 2262.) 

(1215) The ridge-and-furrow system is merely an im- 
proved form of the flooding system, as adapted to specially 
prepared ground. A ditch is run along the top of the arti- 
ficially made ridges, and allowed to overflow along the ad- 
joining slopes. The pipe-and-hydrant system consists in 
running pipes supplied with hydrants somewhat as would be 
done for sprinkling irrigation. The catch-work method is 
virtually the same as the " check" system of irrigation. It 
is adapted to steeper ground than the ridge-and-furrow 
method, and is generally preferred when the ground has a 
sufficient slope to permit of its use. The absorption-ditch 
system has its parallel in the furrow system of irrigation. 
It is particularly well adapted to the application of sewage 
to intermittent filter beds, when it is desired to raise crops 
upon them. (Arts. 2264 to 2268.) 

(1216) Plain water irrigation can and should be prac- 
tised year after year upon the same piece of land. In sew- 
age irrigation, it is frequently found advantageous to divide 
the land into small plots, applying the sewage in rotation to 
each, rather than treating the entire area as a whole. (Art. 
2269.) 

(1217) That the water of all natural streams belongs 
to the public until appropriated to beneficial use, and then 
to the prior appropriator. (Art. 2271.) 



A TEXTBOOK 



ON 



HYDRAULIC ENGINEERING 



International Correspondence Schools 

SCRANTON, PA. 



TABLES AND FORMULAS 



SCRANTON 

INTERNATIONAL TEXTBOOK COMPANY 

A-4 



Copyright, 1897, 1898, by The Colliery Engineer Company. 



All rights reserved. 



TABLES AND FORMULAS. 



This volume contains all the principal Tables and 
Formulas which are likely to be used by the student in 
practice. They have been collected and placed in this 
volume in order to make them convenient for ready refer- 
ence, so that the student will not be obliged to hunt them 
out in the preceding volumes. The number after each 
formula is the same as the number following the same 
formula in the text. 









TABLE 










COMMON 


OF 

LOGARITHMS 








OK NUMBERS 












From 1 to 1 0,000. 








N 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 




i 


M CO 


20 

21 


30 103 
32 222 


40 

41 


60 206 


60 

61 


77 815 


80 

81 


90 309 
90 849 


OO OOO 


61 278 


78 533 


2 


30 IO3 


22 


34 242 


42 


62 325 


62 


79 239 


82 


91 381 


3 


47 712 


23 


36 173 


43 


63 347 


63 


79 934 


83 


91 908 


4 


60 206 


24 


38 021 


44 


64 345 


64 


80 618 


84 


92 428 


5 


69 897 


25 


39 794 


45 


65 321 


65 


81 291 


85 


92 942 


6 


77 815 


26 


41 497 


46 


66 276 


66 


81 954 


86 


93 45o 


7 


84 510 


27 


43 136 


47 


67 210 


67 


82 607 


87 


93 952 


8 


90 309 


28 


44 7i6 


48 


68 124 


68 


83 251 


88 


94 448 


9 
10 

ii 


95 424 


29 
30 

3i 


46 240 


49 
50 

5i 


69 020 


69 

ro 

71 


83 885 


89 
90 
9i 


94 939 


OO OOO 


47 712 


69 897 


84 510 


95 424 


04 139 


49 r 36 


70 757 


85 126 


95 9°4 


12 


07 918 


32 


5o 515 


52 


71 600 


72 


85 733 


92 


9 6 379 


13 


« 394 


33 


5i 851 


53 


72 428 


73 


86 332 


93 


96 S4S 


14 


14 613 


34 


53 148 


54 


73 239 


74 


86 923 


94 


97 3i3 


15 


17 609 


35 


54 407 


55 


74 036 


75 


87 506 


95 


97 77? 


16 


20 412 


36 


55 630 


56 


74 819 


76 


SS 081 


96 


9S 227 


17 


23 045 


37 


56 820 


57 


75 587 


77 


88 649 


97 


9 S 677 


18 


25 527 


38 


57 978 


58 


76 343 


78 


89 209 


9 S 


99 I2 3 


19 
20 


27 875 


39 
40 


59 106 


59 
60 


77 085 


79 

80 


S 9 763 


99 
100 


99 564 


30 103 


60 206 


77 815 


90 309 


OO OOO 



LOGARITHMS. 




LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


150 

151 


17 609 


638 


667 


696 


725 


754 


782 


8n 


840 


869 




898 


926 


955 


984 


*oi3 


*o4i 


*o7o 


*og9 


*I27 


*i56 


152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


441 


29 


28 


153 


469 


498 


526 


554 


583 


611 


639 


667 


696 


724 


1 


2 
5 
8 


•9 
.8 


2 


8 

6 


i54 


752 


780 


808 


837 


865 


893 


921 


949 


977 


*oo5 


3 


•7 


5 
8 


4 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


257 


285 


4 


11 


.6 


n 


2 


156 


312 


340 


368 


396 


424 


451 


479 


507 


535 


562 


5 
6 


T 4 


•5 
•4 
•3 


14 
16 



8 


157 


590 


618 


645 


673 


700 


728 


756 


783 


811 


838 


7 


20 


19 


6 


158 


866 


893 


921 


948 


976 


*oo3 


*03o 


*o58 


*o85 


*II2 


8 


23 


.2 


22 


4 


159 
160 

161 


20 140 


167 


194 


222 


249 


276 


303 


33o 


358 


385 


9 


26.1 


25.2 


412 


439 


466 


493 


520 


548 


575 


602 


629 


656 




683 


710 


737 


763 


790 


817 


844 


871 


898 


925 


162 


952 


978 


*oo5 


*032 


*o5 9 


*o8 5 


*II2 


*I3 9 


*i65 


*ig2 


2 * 


26 


163 


21 219 


245 


272 


299 


325 


352 


378 


405 


43i 


458 


1 


2.7 


2.6 


164 


484 


511 


537 


564 


59° 


617 


643 


669 


696 


722 


2 
3 


5-4 
8.1 


5-2 

7.8 


165 


748 


775 


801 


827 


854 


880 


906 


932 


958 


985 


4 


10.8 


10.4 


166 


22 on 


037 


063 


089 


115 


141 


I6 7 


194 


220 


246 


5 


13-5 
16.2 


13.0 

15.6 
18.2 


167 


272 


298 


324 


35o 


376 


401 


427 


453 


479 


505 


7 


i8!g 


168 


53i 


557 


583 


608 


634 


660 


686 


712 


737 


763 


8 


21.6 


20.8 


169 

iro 

171 


789 


814 


840 


866 


891 


917 


943 


968 


994 


*oig 


9 


24.3 


23.4 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 




300 


325 


35o 


376 


401 


426 


452 


477 


502 


528 


172 


553 


578 


603 


629 


654 


679 


704 


729 


754 


779 


25 


173 


805 


830 


855 


880 


9°5 


930 


955 


980 


*oo5 


*030 


1 


2-5 


174 


24 055 


080 


105 


130 


155 


180 


204 


229 


254 


279 


2 

3 
4 


5-o 
7-5 
10. 


175 


304 


329 


353 


378 


403 


428 


452 


477 


502 


527 


176 


55i 


576 


601 


625 


650 


674 


699 


724 


748 


773 


5 


12.5 


177 


797 


822 


846 


871 


895 


920 


944 


969 


993 


*oi8 


6 


15.0 

17-5 
20.0 


178 


25.042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


8 


179 

180 

181 


285 


310 


334 


358 


382 


406 


431 


455 


479 


503 


Q 


22.5 


527 


55i 


575 


600 


624 


648 


672 


696 


720 


744 




768 


792 


816 


840 


864 


888 


912 


935 


959 


983 


182 


26 007 


031 


o55 


079 


102 


126 


150 


174 


198 


221 


24 


23 


183 


245 


269 


293 


316 


340 


364 


387 


411 


435 


458 


1 


2.4 


2-3 


184 


482 


505 


529 


553 


576 


600 


623 


647 


670 


694 


2 


4.8 


4.6 
6.9 
9.2 


185 


717 


741 


764 


788 


811 


834 


858 


881 


905 


928 


3 
4 


7.2 
9.6 


186 


95i 


975 


998 


*02I 


*o45 


*o68 


*ogi 


*H4 


*I38 


*i6i 


5 


12.0 


"•5 


187 


27 184 


207 


231 


254 


277 


300 


323 


346 


37o 


393 


6 


14.4 
16.8 
19.2 


13.8 

16. 1 

18.4 


188 


416 


439 


462 


485 


508 


53i 


554 


577 


600 


623 


7 
8 


189 
190 

191 


646 


669 


692 


715 


738 


761 


784 


807 


830 


852 


9 


21.6 


20.7 


875 


898 


921 


944 


967 


989 


*OI2 


*Q35 


*Q58 


*o8i 




28 103 


126 


149 


171 


194 


217 


240 


262 


285 


307 


192 


33o 


353 


375 


398 


421 


443 


466 


488 


5ii 


533 


11 


21 


193 


556 


578 


601 


623 


646 


668 


69I 


713 


735 


758 




2.2 


2.1 


194 


780 


803 


825 


847 


870 


892 


914 


937 


959 


981 


2 


4.4 


4 


2 


195 


29 003 


026 


048 


070 


092 


115 


137 


159 


181 


203 


3 


6.6 

S.8 


6 
8 


3 


196 


226 


248 


270 


292 


3U 


336 


358 


3S0 


403 


425 


s 


11. 


10 


5 


197 


447 


469 


491 


5i3 


535 


557 


579 


601 


623 


645 


6 


13.2 


12 


6 


198 


667 


688 


710 


732 


754 


776 


798 


820 


842 


863 


7 

s 


i5-4 
17.6 
19.8 


14 

16 


7 

s 


199 
200 


885 


907 


929 


95i 


973 


994 


*oi6 


*o 3 S 


*o6o 


*o8i 


9 


iS 


9 


30 103 


125 


146 


168 


190 


211 


233 


255 


276 


.298 




N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


200 

20 1 


30 103 


125 


146 


168 


190 


211 


233 


255 


276 


298 




320 


341 


363 


384 


406 


428 


449 


471 


492 


514 


202 


535 


557 


578 


600 


621 


643 


664 


685 


707 


728 




21 

2 .1 


203 


75o 


771 


792 


814 


835 


856 


878 


899 


920 


942 


1 


2.2 


204 


9 6 3 


984 


*oo6 


*027 


*o 4 8 


*o6g 


*c>9i 


*II2 


*I33 


*I54 


2 


4.4 


4.2 


205 


3i 175 


197 


218 


239 


260 


281 


302 


323 


345 


366 


3 


6.6 
8.8 
11. 


i- 3 

8.4 

10.5 


206 


387 


408 


429 


450 


47i 


492 


513 


534 


555 


576 


4 
5 


207 


597 


618 


639 


660 


681 


702 


723 


744 


765 


785 


6 


13.2 


12.6 


208 


806 


827 


848 


869 


890 


911 


93i 


952 


973 


994 


7 
8 
9 


15-4 
17.6 
19. 8 


14.7 
16.8 
18.9 


209 
210 

211 


32 015 


035 


056 


077 


098 


118 


139 


160 


181 


201 


222 


243 


263 


284 


305 


325 


346 


366 


387 


408 




428 


449 


469 


490 


5io 


531 


552 


572 


593 


613 


212 


634 


654 


675 


695 


7i5 


736 


756 


777 


797 


818 




20 


213 


838 


858 


879 


899 


919 


940 


960 


980 


*OOI 


*02I 


1 


2.0 


214 


33 041 


062 


082 


102 


122 


143 


163 


183 


203 


224 


2 

3 


4.0 
6.0 


215 


244 


264 


284 


304 


325 


345 


365 


385 


405 


425 


4 


8.0 


216 


445 


465 


486 


506 


526 


546 


566 


586 


606 


626 


I 

7 


10. 


217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


826 


12 .0 
14.0 


218 


846 


866 


885 


9° 5 


925 


945 


965 


985 


*oo5 


*025 


8 


16.0 


219 
220 

221 


34 044 


064 


084 


104 


124 


143 


163 


183 


203 


223 


9 


18.0 


242 


262 


282 


301 


321 


34i 


361 


380 


400 


420 


<a 


439 


459 


479 


498 


5i8 


537 


557 


577 


596 


616 


222 


635 


655 


674 


694 


713 


733 


753 


772 


792 


811 


t 


1.9 


223 


830 


850 


869 


889 


908 


928 


947 


967 


986 


*oo5 


2 


3-8 


224 


35 025 


044 


064 


083 


102 


122 


141 


160 


180 


199 


3 


7.6 
9-5 


225 


218 


238 


257 


276 


295 


315 


334 


353 


372 


392 


4 

5 


226 


411 


43o 


449 


468 


488 


507 


526 


545 


564 


583' 


6 


11. 4 


227 


603 


622 


641 


660 


679 


698 


717 


736 


755 


774 


7 
8 
9 


13-3 


228 


793 


813 


832 


851 


870 


889 


908 


927 


946 


965 


15.2 

17. 1 


229 
230 

231 


984 


*oo3 


*02I 


*040 


*o5 9 


*o78 


*o 9 7 


*n6 


*i35 


*i54 


18 


36 173 


192 


211 


229 


248 


267 


286 


305 


324 


342 


361 


380 


399 


418 


430 


455 


474 


493 


5ii 


53o 


232 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


1 


1.8 

n ft 


233 


736 


754 


773 


791 


810 


829 


847 


866 


884 


9°3 


3 


3-° 
5-4 


234 


922 


940 


959 


977 


996 


*oi4 


*Q33 


*o5i 


*o7o 


*o88 


4 


7.2 


235 


37 107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


5 


9.0 
10.8 


236 


291 


310 


328 


346 


365 


383 


401 


420 


438 


457 


7 


12.6 


237 


475 


493 


5ii 


530 


548 


566 


585 


603 


621 


639 


8 


14.4 


238 


658 


676 


694 


712 


73i 


749 


767 


785 


803 


822 


9 


16.2 


239 
240 
241 


840 


858 


876 


894 


912 


93i 


949 


967 


985 


*oc>3 
184 




38 021 


039 


057 


o75 


093 


112 


130 


148 


166 


202 


220 


238 


256 


274 


292 


310 


328 


346 


364 


I 


1Y 

'•7 


242 


382 


399 


417 


435 


453 


471 


489 


507 


525 


543 


2 


3-4 


243 


56i 


578 


596 


614 


632 


650 


668 


686 


703 


721 


3 


5-i 

6 8 


244 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 


4 
5 


8! 5 


245 


917 


934 


952 


970 


987 


*oo5 


*023 


*o4i 


*o58 


*o76 


6 


10.2 


246 


39 094 


in 


129 


146 


164 


182 


199 


217 


235 


252 


7 
8 


11. 9 
n 6 


247 


270 


287 


305 


322 


34o 


358 


375 


393 


410 


428 


9 


IS- 3 


248 


445 


463 


480 


498 


5i5 


533 


55o 


568 


585 


602 




249 
250 


620 


637 


655 


672 


690 


707 


724 


-742 


759 


777 
95o 




794 


811 


829 


846 


863 


881 


898 


915 


933 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. 


P. 


250 

251 


39 794 


811 


829 


846 


863 
*037 


881 


898 


915 


933 


950 






967 


985 


*002 


*oi9 


*o54 


*o7i 


*o88 


*io6 


*I23 


252 


40 140 


157 


175 


192 


209 


226 


243 


261 


278 


295 




18 


253 


312 


329 


346 


364 


381 


398 


415 


432 


449 


466 


1 


1.8 


254 


483 


500 


518 


535 


552 


569 


586 


603 


620 


637 


2 


3-6 


255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


3 
4 


5-4 
7.2 


256 


824 


841 


858 


875 


892 


909 


926 


943 


960 


976 


5 


9.0 


257 


993 


*OIO 


*027 


*o44 


*o6i 


*o78 


*o 9 5 


*iii 


*I28 


*i45 


6 


10.8 


258 


41 162 


179 


196 


212 


229 


246 


263 


280 


296 


313 


7 
8 


12.6 
14.4 


259 
260 

261 


33o 


347 


363 


380 


397 


414 


430 


447 


464 


481 


9 


16.3 


497 


514 


531 


547 


564 


58i 


597 


614 


631 


647 


664 


681 


697 


714 


731 


747 


764 


780 


797 


814 


262 


830 


847 


863 


880 


896 


9 T 3 


929 


946 


9^3 


979 




17 


263 


996 


*OI2 


*029 


*Q45 


*o62 


*o78 


*095 


*m 


*I27 


*I44 


1 


1-7 


264 


42 160 


177 


193 


210 


226 


243 


259 


275 


292 


308 


2 


3-4 


265 


325 


341 


357 


374 


39° 


406 


423 


439 


455 


472 


3 
4 


5- 1 

6.8 


266 


488 


504 


521 


537 


553 


57o 


586 


602 


619 


635 


5 


8-5 


267 


651 


667 


684 


700 


716 


732 


749 


765 


78i 


797 


6 


10.2 


268 


813 


83O 


846 


862 


878 


894 


.911 


927 


943 


959 


7 
8 


11. 9 
13.6 
15.3 


269 

2ro 

271 


975 


991 


*oo8 


*024 


*040 


*o56 


'• 072 


*o88 


*io4 


*I20 


9 


43 136 


152 


169 


185 


201 


217 


233 


249 


265 


28l 


297 


313 


329 


345 


361 


377 


393 


409 


425 


441 


272 


457 


473 


489 


505 


521 


537 


553 


569 


584 


600 




16 


273 


616 


632 


648 


664 


680 


696 


712 


727 


743 


759 


1 


1.6 


274 


775 


791 


807 


823 


838 


854 


870 


886 


902 


917 


2 


3-2 

4.8 
6.4 


275 


933 


949 


965 


981 


996 


*OI2 


*028 


*o44 


*Q59 


*o 7 5 


3 
4 


276 


44 091 


107 


122 


138 


154 


I70 


185 


201 


217 


232 


I 


8.0 


277 


248 


264 


279 


295 


3ii 


326 


342 


358 


373 


389 


9 .6 


278 


404 


420 


436 


451 


467 


483 


498 


514 


529 


545 


7 

8 


11 .2 

12.8 


279 

280 

281 


560 


576 


592 


607 


623 


638 


654 


669 


685 


700 


9 


14.4 


716 


73i 


747 


762 


778 


793 


809 


824 


840 


855 


871 


886 


902 


917 


932 


948 


963 


979 


994 


*OIO 


282 


45 025 


040 


056 


071 


086 


102 


117 


133 


148 


163 




15 


283 


179 


194 


209 


225 


240 


255 


271 


286 


301 


3i7 


X 


i-5 


284 


332 


347 


362 


378 


393 


408 


423 


439 


454 


469 


a 


3-o 


285 


484 


500 


515 


53o 


545 


56i 


576 


591 


606 


621 


3 
4 


4-5 
6.0 


286 


637 


652 


667 


682 


697 


712 


728 


743 


758 


773 


5 


7-5 


287 


788 


803 


818 


834 


849 


864 


879 


894 


909 


924 


6 


9.0 


288 


939 


954 


969 


984 


*ooo 


*oi5 


*030 


*o 4 5 


*o6o 


*Q75 


7 
8 


10.5 
12.0 


289 
290 

291 


46 090 


105 


120 


135 


150 


165 


180 


195 


210 


225 


9 


13.5 


240 


255 


270 


285 


300 


3i5 


330 


345 


359 


374 


389 


404 


419 


434 


449 


464 


479 


494 


509 


523 


292 


538 


553 


568 


583 


598 


613 


627 


642 


657 


672 




14 


293 


687 


702 


716 


731 


746 


761 


776 


790 


805 


820 


X 


1.4 


294 


835 


850 


864 


879 


894 


909 


923 


938 


953 


967 


3 


2.8 


295 


982 


997 


*OI2 


*026 


*04i 


*o56 


*o7o 


*o85 


*IOO 


*ii4 


3 
4 
5 


4.2 
5-6 
7.0 


296 


47 129 


144 


159 


173 


188 


202 


217 


232 


246 


261 


297 


276 


290 


305 


319 


334 


349 


363 


378 


392 


407 


6 


8.4 


298 


422 


436 


451 


465 


480 


494 


509 


524 


538 


553 


7 
g 


9.8 
11. 2 


299 
300 


567 


582 


596 


611 


625 


640 


654 


669 


683 


698 


9 


13.6 


712 


727 


741 


756 


770 


784 


799 


813 


828 


842 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. 


P. 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


300 

301 


47 712 


727 


741 


756 


770 


784 


799 


813 


828 


842 




857 


871 


885 


900 


914 


929 


943 


958 


972 


986 


302 


48 001 


015 


029 


044 


058 


073 


087 


IOI 


116 


130 




303 


144 


159 


173 


187 


202 


216 


230 


244 


259 


273 




304 


287 


302 


316 


330 


344 


359 


373 


387 


401 


416 




305 


430 


444 


458 


473 


487 


501 


515 


53o 


544 


558 


f 


■0 
x-5 

3-° 


306 


572 


536 


601 


615 


629 


643 


657 


671 


686 


700 


2 


307 


714 


728 


742 


756 


770 


785 


799 


813 


827 


841 


3 


4-5 


308 


855 


869 


,883 


897 


911 


926 


940 


954 


968 


982 


4 

5 

6 
7 
8 

9 


6.0 

7-5 
9.0 
10.5 
12.0 
13-5 


309 
310 

311 


996 


*OIO 


*024 


*o3S 


*052 


■•066 


*o8o 


*o 9 4 


*io8 


*I22 


49 136 


150 


164 


178 


192 


206 


220 


234 


248 


262 


276 


290 


304 


318 


332 


346 


360 


374 


388 


402 


312 


415 


429 


443 


457 


471 


485 


499 


513 


527 


541 




3i3 


554 


568 


582 


596 


610 


624 


638 


651 


665 


679 




3U 


693 


707 


721 


734 


748 


762 


776 


790 


803 


817 




3i5 


831 


845 


859 


872 


886 


900 


914 


927 


941 


955 




316 


969 


982 


996 


*OIO 


*024 


-037 


*o5i 


*o65 


*o79 


*og2 


14 


3i7 


50 106 


120 


133 


147 


161 


174 


188 


202 


215 


229 


1 
2 


j.. 4 
2.8 


318 


243 


256 


270 


284 


297 


311 


325 


333 


352 


365 


3 


4.2 


319 
320 

321 


379 


393 


406 


420 


433 


447 


461 


474 


488 


501 


4 

5 
6 
7 
8 


5-6 
7.0 

8.4 
9.8 
11. 2 


515 


529 


542 


556 


569 


583 


596 


610 


623 


637 


651 


664 


678 


691 


705 


718 


732 


745 


759 


772 


322 


786 


799 


813 


826 


840 


853 


866 


880 


893 


907 


9 


12.6 


323 


920 


934 


947 


961 


974 


987 


*OOI 


*oi4 


*028 


*04i 




324 


5i 055 


068 


081 


095 


108 


121 


135 


148 


162 


175 




325 


188 


202 


215 


228 


242 


255 


268 


282 


295 


308 




326 


322 


335 


348 


362 


375 


388 


402 


415 


428 


441 




327 


455 


468 


481 


495 


508 


521 


534 


548 


56i 


574 


13 


328 


587 


601 


614 


627 


640 


654 


667 


680 


693 


706 


1 


1-3 
2 .6 


329 
330 

33i 


720 


733 


746 


759 


772 


786 


799 


812 


825 


838 


3 
4 
5 
6 


3-9 

5-2 

6-5 
7.8 


851 


865 


878 


891 


904 


917 


930 


943 


957 


970 


983 


996 


*oo9 


*022 


*Q35 


*o48 


*o6i 


*Q75 


*o88 


*IOI 


332 


52 114 


127 


140 


153 


166 


179 


192 


205 


218 


231 


7 

8 


9.1 
10.4 
11,7 


333 


244 


257 


270 


284 


297 


310 


323 


336 


349 


362 


9 


334 


375 


388 


401 


414 


427 


440 


453 


466 


479 


492 




335 


504 


517 


530 


543 


556 


569 


582 


595 


608 


621 




336 


634 


647 


660 


673 


686 


699 


711 


724 


737 


75o 




337 


763 


776 


789 


802 


815 


827 


840 


853 


866 


879 




338 


892 


9°5 


917 


930 


943 


956 


969 


982 


994 


*oo7 


12 


339 
340 

34i 


53 020 


033 


046 


058 


071 


084 


097 


no 


122 


135 


1 
2 
3 
4 
5 


1.2 

2.4 
3-6 
4.8 
6.0 


148 


161 


173 


186 


199 


212 


224 


237 


250 


263 


275 


288 


301 


314 


326 


339 


352 


364 


377 


390 


342 


403 


415 


428 


441 


453 


466 


479 


49 1 


504 


517 


6 


7.2 


343 


529 


542 


555 


567 


580 


593 


605 


618 


631 


643 


7 
8 
g 


8.4 
9.6 
10.8 


344 


656 


668 


681 


694 


706 


719 


732 


744 


757 


769 


345 


782 


794 


807 


820 


832 


845 


857 


870 


882 


895 




346 


908 


920 


933 


945 


958 


970 


933 


995 


*oo8 


*020 




347 


54 033 


045 


058 


070 


083 


095 


108 


120 


133 


145 




343 


158 


170 


183 


195 


208 


220 


233 


245 


258 


270 




349 
350 


283 


295 


307 


320 


332 


345 


357 


37o 


382 


394 




407 


419 


432 


444 


456 


469 


4S1 


494 


506 


518 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


350 

35i 


54 407 


419 


432 


444 


456 


469 


481 


494 


506 


518 




531 


543 


555 


568 


580 


593 


605 


617 


630 


642 


352 


654 


667 


679 


691 


704 


716 


728 


741 


753 


765 




353 


777 


790 


802 


814 


827 


839 


851 


864 


876 


888 




354 


900 


9*3 


925 


937 


949 


962 


974 


986 


998 


*OII 


13 


355 


55 023 


035 


047 


060 


072 


084 


096 


108 


121 


133 


X 


i-3 
2.6 


356 


145 


157 


169 


182 


194 


206 


218 


230 


242 


255 


3 


3-9 


357 


267 


279 


291 


303 


315 


328 


34o 


352 


364 


376 


4 


5-2 


358 


388 


400 


4i3 


425 


437 


449 


461 


473 


485 


497 


5 
6 


6-5 
7 8 


359 
360 

361 


509 


522 


534 


546 


558 


57o 


582 


594 


606 


618 


7 
8 
9 


/ - u 
9.1 
10.4 
11 .n 


630 


642 


654 


666 


678 


691 


703 


715 


727 


739 


75i 


763 


775 


787 


799 


811 


823 


835 


847 


859 




362 


871 


883 


895 


907 


919 


,931 


943 


955 


967 


979 




363 


991 


*oo3 


*oi5 


*027 


*o 3 8 


*o5o 


*o62 


*o74 


*o86 


*o 9 8 




364 


56 no 


122 


134 


146 


158 


170 


182 


194 


205 


217 




365 


229 


241 


253 


265 


277 


289 


301 


312 


324 


336 




366 


348 


360 


372 


384 


39 6 


407 


419 


43i 


443 


455 


12 


367 


467 


478 


490 


502 


514 


526 


538 


549 


561 


573 


1 


1.2 


368 


585 


597 


608 


620 


632 


644 


656 


667 


679 


691 


3 


2.4 

3-6 


369 
3?0 

37i 


703 


714 


726 


738 


75o 


761 


773 


785 


797 


808 


4 
5 
6 

7 
8 


4.8 

6.0 
7.2 
8.4 
9.6 
10.8 


820 


832 


844 


855 


867 


879 


891 


902 


914 


926 


937 


949 


961 


972 


984 


996 


*oo8 


*oi9 


*03i 


*<H3 


372 


57 054 


066 


078 


089 


IOI 


113 


124 


136 


148 


159 


9 


373 


171 


183 


194 


206 


217 


229 


241 


252 


264 


276 




374 


287 


299 


310 


322 


334 


345 


• 357 


368 


380 


392 




375 


403 


415 


426 


438 


449 


461 


473 


484 


496 


507 




376 


519 


530 


542 


553 


565 


576 


588 


600 


611 


623 




377 


634 


646 


657 


669 


680 


692 


703 


715 


726 


738 




378 


749 


761 


772 


784 


795 


807 


818 


830 


841 


852 




11 


379 
580 

381 


864 


875 


887 


898 


910 


921 


933 


944 


955 


967 


2 

3 
4 
5 


2.2 

3-3 
4.4 


978 
58 092 


990 


*OOI 


*oi3 


*024 


*035 


*Q47 


*o 5 8 


*070 


*o8i 


104 


115 


127 


138 


149 


161 


172 


184 


195 


382 


206 


218 


229 


240 


252 


263 


274 


286 


297 


309 


6 
7 
8 


6.6 
7-7 
8.8 


383 


320 


331 


343 


354 


365 


377 


388 


399 


410 


422 


384 


433 


444 


456 


467 


478 


490 


501 


512 


524 


535 


9 


9.9 


385 


546 


557 


569 


580 


591 


602 


614 


625 


636 


647 




386 


659 


670 


681 


692 


704 


715 


726 


737 


749 


760 




387 


771 


782 


794 


805 


816 


827 


838 


850 


861 


872 




388 


883 


894 


906 


917 


928 


939 


95o 


961 


973 


984 




389 
390 

39 1 


995 


*oo6 


*oi7 


*028 


^040 


*o5i 


*o62 


*o73 


*o8 4 


*o 9 5 


<n 


59 106 


118 


129 


140 


151 


162 


173 


184 


195 


207 


2 
3 


1.0 

2.0 
3.0 


218 


229 


240 


251 


262 


273 


284 


295 


306 


3i8 


392 


329 


340 


35i 


362 


373 


384 


395 


406 


417 


428 


4 


4.0 


393 


439 


45o 


461 


472 


483 


494 


506 


517 


528 


539 


5 
6 
7 


6.0 
7.0 


394 


55o 


56i 


572 


583 


594 


605 


616 


627 


638 


649 


395 


660 


671 


682 


693 


704 


715 


726 


737 


748 


759 


8 


8.0 


39 6 


770 


780 


791 


802 


813 


824 


835 


846 


857 


868 





9.0 


397 


879 


890 


901 


912 


923 


934 


945 


956 


966 


977 




398 


988 


999 


*OIO 


*02I 


*032 


*043 


*Q54 


*o65 


*o76 


*oS6 




399 
400 


60 097 


108 


119 


I30 


141 


152 


163 


173 


184 


195 




206 


217 


228 


239 


249 


260 


271 


2S2 


293 


304 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



O. O. IV.— 23 



LOGARITHM& 



N. 


L. 1 1 


•■ 


3 1 • 


5 


6 | 7 


8 


9 


P. P. 


400 

401 


Go 206 


217 


223 


239 


249 


260 


271 282 


293 


304 




314 


325 


336 


347 


353 


369 


379 


390 


401 


412 


402 


423 


433 


444 


455 


466 


477 


4S7 


498 


509 


520 




403 


531 


541 


552 


563 


574 


5S4 


595 


606 


617 


627 




404 


638 


649 


660 


670 


6S1 


692 


703 


713 


724 


735 




405 


746 


756 


767 


778 


788 


799 


810 


821 


831 


842 




406 


853 


S63 


874 


8S5 


895 


906 


917 


927 


938 


949 


11 


407 


959 


970 


981 


991 


*002 


*oi 3 


"02 3 


*034 


*Q45 


*o55 


1 


1.1 


40S 


61 066 


077 


087 


098 


109 


119 


I30 


140 


151 


162 


2 


2.2 


409 
410 

411 


172 


183 


194 


204 


215 


225 


236 


247 


257 


268 


3 
4 
5 

6 
7 


3-3 
4.4 
5-5 
6.6 
7-7 


278 


2S9 


300 


310 


321 


33i 


342 


352 


363 


374 


3S4 


395 


405 


416 


426 


437 


443 


458 


469 


479 


412 


490 


500 


5ii 


521 


532 


542 


553 


563 


574 


584 


8 


8.8 


413 


595 


606 


616 


627 


637 


648 


658 


669 


679 


690 


9 


9.9 


4U 


700 


711 


721 


73i 


742 


752 


763 


773 


784 


794 




415 


805 


815 


826 


836 


847 


857 


863 


878 


8S8 


899 




416 


909 


920 


930 


94i 


951 


962 


972 


982 


993 


*oo3 




417 


62 014 


024 


034 


045 


055 


066 


076 


o36 


097 


107 




41S 


118 


128 


13S 


149 


159 


170 


1S0 


190 


201 


211 




419 
420 

421 


221 


232 


242 


252 


263 


273 


284 


294 


304 


315 


«n 


325 


335 


346 


356 


366 


377 


337 


397 


40S 


418 


428 


439 


449 


459 


469 


480 


490 


500 


5ii 


521 


422 


53i 


542 


552 


562 


572 


533 


593 


603 


613 


624 


1 


1.0 


423 


634 


644 


655 


665 


675 


685 


696 


706 


716 


726 


2 


2.0 


424 


737 


747 


757 


767 


778 


7S3 


798 


S08 


818 


829 


3 


3-° 


425 


839 


849 


859 


870 


880 


890 


900 


910 


921 


931 


4 
5 


4.0 
5-o 


426 


94i 


95i 


961 


972 


9S2 


992 


*002 


*OI2 


*022 


*033 


6 


6.0 


427 


63 043 


053 


063 


073 


0S3 


094 


IO4 


114 


124 


134 


7 
g 


7.0 
8.0 


428 


144 


155 


165 


175 


185 


195 


205 


215 


225 


236 


g 


Q.O 


429 
430 

43i 


246 


256 


266 


276 


286 


296 


306 


317 


327 


337 




347 


357 


367 


377 


3S7 


397 


407 


417 


42S 


433 


443 


45S 


468 


478 


4S8 


498 


508 


518 


528 


538 


432 


548 


558 


568 


579 


589 


599 


609 


619 


629 


639 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 




434 


749 


759 


769 


779 


789 


799 


809 


8T9 


829 


839 




435 


849 


859 


869 


879 


889 


899 


909 


9I9 


929 


939 




436 


949 


959 


969 


979 


9S8 


998 


*oo8 


*oi8 


*023 


*038 




437 


64 048 


058 


068 


07S 


088 


098 


108 


11S 


128 


137 


O 


433 


147 


157 


167 


177 


IS 7 


197 


207 


217 


227 


237 


1 


O.g 


439 
440 

44i 


246 


256 


266 


276 


2S6 


296 


306 


316 


326 


335 


2 
3 
4 
5 
6 


l.S 

2.7 

3-6 
4-5 
5-4 
6-3 


345 


355 


365 


375 


3S5 


395 


404 


414 


424 


434 


444 


454 


464 


473 


4S3 


493 


503 


513 


523 


532 


442 


542 


552 


562 


572 


582 


591 


601 


611 


621 


631 


7 


443 


640 


650 


660 


670 


6S0 


6S9 


699 


709 


719 


729 


8 


7.2 


444 


738 


748 


758 


76S 


777 


737 


797 


807 


816 


826 


9 


8.1 


445 


836 


846 


856 


865 


875 


8S5 


895 


904 


914 


924 




446 


933 


943 


953 


9 6 3 


972 


982 


992 


*002 


"Oil 


*02I 




447 


65 031 


040 


050 


060 


070 


079 


089 


O99 


108 


Il8 




44S 


128 


137 


147 


157 


167 


176 


186 


I96 


205 


215 




449 
450 


225 


234 


244 


254 


263 


273 


283 


292 


302 


312 




321 


33i 


34i 


35o 


360 


369 


379 


389 


393 


408 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



N. 


L. 1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


450 

45i 


65 321 


331 


341 


350 


360 


369 


379 


389 


398 


408 




418 


427 


437 


447 


456 


466 


475 


485 


495 


504 


452 


514 


523 


533 


543 


552 


562 


571 


581 


59i 


600 




453 


610 


619 


629 


639 


648 


658 


667 


677 


686 


696 




454 


706 


715 


725 


734 


744 


753 


763 


772 


782 


792 




455 


801 


811 


820 


830 


839 


849 


858 


868 


877 


887 




456 


896 


906 


916 


9 2 5 


935 


944 


954 


9 6 3 


973 


982 


10 


457 


992 


*OOI 


*OII 


*020 


*03o 


*c>39 


*04 9 


*o58 


*o68 


*o77 


1 


1.0 


458 


66 087 


096 


106 


115 


124 


134 


143 


153 


162 


172 


2 
3 

4 
5 

6 
7 


2.0 
3-o 
4.0 
5-o 
6.0 
7.0 


459 
460 

461 


181 


191 


200 


2IO 


219 


229 


238 


247 


257 


266 


276 


285 


295 


304 


314 


323 


332 


342 


351 


361 


37o 


380 


389 


398 


408 


417 


427 


436 


445 


455 


462 


464 


474 


483 


492 


502 


5ii 


521 


530 


539 


549 


8 

Q 


8.0 


463 


558 


567 


577 


586 


596 


605 


614 


624 


633 


642 




464 


652 


661 


671 


680 


689 


699 


708 


717 


727 


736 




465 


745 


755 


764 


773 


783 


792 


801 


811 


820 


829 




466 


839 


848 


857 


867 


876 


885 


894 


904 


913 


J 22 




467 


932 


941 


95o 


960 


969 


978 


987 


997 


*oo6 


*oi5 




468 


67 025 


034 


043 


052 


062 


071 


080 


089 


099 


108 




469 

4ro 

471 


117 


127 


136 


145 


154 


164 


173 


182 


191 


201 




210 


219 


228 


237 


247 


256 


265 


274 


284 


293 


302 


3ii 


321 


33o 


339 


348 


357 


367 


376 


385 


472 


394 


403 


4i3 


422 


43i 


440 


449 


459 


468 


477 


I 


7 
o.g 


473 


486 


495 


504 


514 


523 


532 


541 


55o 


560 


569 


2 


1.8 


474 


578 


587 


596 


605 


614 


624 


633 


642 


651 


660 


3 


2.7 


475 


669 


679 


688 


697 


706 


715 


724 


733 


742 


752 


4 

5 
6 


3-6 


476 


761 


770 


779 


788 


797 


806 


815 


825 


834 


843 


4*5 
5-4 


477 


852 


861 


870 


879 


888 


897 


906 


916 


925 


934 


7 


6-3 


478 


943 


952 


961 


976 


979 


988 


997 


*oo6 


*oi5 


*024 


8 
g 


7.2 

8.1 


479 

480 

481 


68 034 


043 


052 


061 


070 


079 


088 


097 


106 


115 




124 


133 


142 


151 


160 


169 


178 


187 


196 


205 


215 


224 


233 


242 


251 


260 


269 


278 


287 


296 


482 


305 


314 


323 


332 


34i 


35o 


359 


368 


377 


386 




483 


395 


404 


413 


422 


43i 


440 


449 


458 


467 


476 




484 


485 


494 


502 


5ii 


520 


529 


538 


547 


556 


565 




485 


574 


583. 


59 2 


601 


610 


619 


628 


637 


646 


655 




486 


664 


673 


681 


690 


699 


708 


717 


726 


735 


744 




487 


753 


762 


771 


780 


789 


797 


806 


815 


824 


833 


I 




0.8 


488 


842 


851 


860 


869 


878 


886 


895 


904 


9!3 


922 


2 


1.6 


489 
490 
491 


93i 


940 


949 


958 


966 


975 


984 


993 


*002 


*OII 


3 
4 
5 
6 
7 


2.4 

3-2 

4.0 
4.8 
5.6 


69 020 


028 


037 


046 


o55 


064 


073 


082 


O9O 


099 


108 


117 


126 


135 


144 


152 


161 


170 


179 


188 


492 


197 


205 


214 


223 


232 


241 


249 


258 


267 


276 


8 


6.4 


493 


285 


294 


302 


3ii 


320 


329 


338 


346 


355 


364 


9 


7.3 


494 


373 


38i 


390 


399 


408 


417 


425 


434 


443 


452 




495 


461 


469 


478 


487 


496 


504 


513 


522 


531 


539 




496 


548 


557 


566 


574 


583 


592 


601 


609 


618 


627 




497 


636 


644 


653 


662 


671 


679 


688 


697 


705 


714 




498 


723 


732 


740 


749 


758 


767 


775 


784 


793 


801 




499 
500 


810 


819 


827 


836 


845 


854 


862 


871 


880 


88S 




897 


906 


914 


923 


932 


940 


949 


958 


966 


975 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 8 


9 


P. P. 



10 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


500 

501 


69 897 


906 


914 


923 


932 


940 


949 


958 


966 


975 




984 


992 


*OOI 


*OIO 


*oi8 


*027 


*c>36 


*044 


*Q53 


*o62 


502 


70 070 


079 


088 


096 


105 


114 


122 


131 


140 


148 




503 


157 


165 


174 


183 


191 


200 


209 


217 


226 


234 




504 


243 


252 


260 


269 


278 


286 


295 


303 


312 


321 




505 


329 


338 


346 


355 


364 


372 


38i 


389 


398 


406 




506 


415 


424 


432 


441 


449 


458 


467 


475 


484 


492 




507 


501 


509 


518 


526 


535 


544 


552 


56i 


569 


578 


9 


508 


586 


595 


603 


612 


621 


629 


638 


646 


655 


663 


1 


O.Q 

1.8 


509 
510 

5ii 


672 


680 


689 


697 


706 


714 


723 


73i 


740 


749 


3 
4 
5 

6 


2.7 

3-6 
4-5 
5-4 


757 


766 


774 


783 


791 


800 


808 


817 


825 


834 


842 


851 


859 


868 


876 


885 


893 


902 


910 


919 


512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*oc>3 


7 
8 


6.3 
7.2 


513 


71 012 


020 


029 


037 


046 


054 


063 


071 


079 


088 


9 


8.1 


514 


096 


105 


113 


122 


130 


139 


147 


155 


164 


172 




515 


181 


189 


198 


206 


214 


223 


231 


240 


248 


257 




5i6 


265 


273 


282 


290 


299 


307 


315 


324 


332 


341 




517 


349 


357 


366 


374 


383 


391 


399 


408 


416 


425 




5i8 


433 


441 


45o 


458 


466 


475 


483 


492 


500 


508 




519 
520 

521 


517 


525 


533 


542 


55o 


559 


567 


575 


584 


592 




600 


609 


617 


625 


634 


642 


650 


659 


667 


675 


684 


692 


700 


709 


717 


725 


734 


742 


75o 


759 


522 


767 


775 


784 


792 


800 


809 


817 


825 


834 


842 


8 

! ^ a 


523 


850 


858 


867 


875 


8S3 


892 


900 


908 


917 


925 


2 


1 


6 


524 


933 


941 


95o 


958 


966 


975 


983 


991 


999 


*oo8 


3 


2 


4 


525 


72 016 


024 


032 


041 


049 


o57 


066 


074 


082 


090 


4 


3 


2 


526 


099 


107 


115 


123 


132 


140 


148 


156 


165 


173 


5 
6 


4 

4 


8 


527 


181 


189 


198 


206 


214 


222 


230 


239 


247 


255 


7 


5 


6 


528 


263 


272 


280 


288 


296 


304 


313 


321 


329 


337 


8 


6 


4 


529 
530 

53i 


346 


354 


362 


37o 


378 


387 


395 


403 


411 


419 


9 


7-* 


428 


436 


444 


452 


460 


469 


477 


485-- 


493 


501 




509 


5i8 


526 


534 


542 


55o 


558 


567 


575 


583 


532 


591 


599 


607 


616 


624 


632 


640 


648 


656 


665 




533 


673 


681 


689 


697 


705 


713 


722 


730 


738 


746 




534 


754 


762 


770 


779 


787 


795 


803 


811 


819 


827 




535 


835 


843 


852 


860 


868 


876 


884 


892 


900 


908 




536 


916 


,925 


933 


941 


949 


957 


9^5 


973 


981 


989 




537 


997 


"006 


^014 


*022 


*030 


"038 


^046 


*©54 


*o62 


-070 


7 


533 


73 078 


086 


094 


I02 


in 


119 


127 


135 


143 


151 


1 


0.7 


539 
540 

54i 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 


3 
4 
5 
6 


1.4 
2.1 

2.8 
3-5 
4.2 


239 


247 


255 


263 


272 


280 


288 


296 


304 


312 


320 


328 


336 


344 


352 


360 


368 


376 


384 


392 


542 


400 


408 


416 


.424 


432 


440 


448 


456 


464 


472 


I 


4.9 
e 6 


543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


9 


1 
6.3 


544 


560 


568 


576 


584 


592 


600 


608 


616 


624 


632 




545 


640 


648 


656 


664 


672 


679 


687 


695 


703 


711 




546 


719 


72 7 


735 


743 


75i 


759 


767 


775 


783 


791 




547 


799 


807 


815 


823 


830 


838 


846 


854 


862 


870 




548 


878 


886 


894 


902 


910 


918 


926 


933 


941 


949 




549 
550 


957 


965 


973 


981 


989 


997 


*oo5 


*oi3 


*020 


*028 




74 036 


044 


052 


060 


068 


076 


084 


092 


O99 


107 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



11 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


550 

55i 


74 036 


044 


052, 


060 


068 


076 


084 


092 


099 


107 




115 


123 


131 


139 


147 


155 


162 


170 


178 


186 


552 


194 


202 


2IO 


218 


225 


233 


241 


249 


257 


265 




553 


273 


280 


288 


296 


304 


312 


320 


327 


335 


343 




554 


351 


359 


367 


374 


382 


39° 


398 


406 


414 


421 




555 


429 


437 


445 


453 


461 


468 


476 


484 


492 


500 




556 


507 


5i5 


523 


53i 


539 


547 


554 


562 


57o 


578 




557 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 




558 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 




559 
560 

561 


741 


749 


757 


764 


772 


780 


788 


796 


803 


811 


8 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 


896 


904 


912 


920 


927 


935 


943 


950 


958 


966 


1 


0.8 


562 


974 


981 


989 


997 


*oo5 


*OI2 


*020 


*028 


*Q35 


*o43 


2 
3 
4 


1.6 

2.4 

3-2 


563 


75 051 


059 


066 


074 


082 


089 


O97 


105 


113 


j. 20 


564 


128 


136 


143 


151 


159 


166 


174 


182 


189 


197 


5 


4.0 


565 


205 


213 


220 


228 


236 


243 


251 


259 


266 


274 


6 

7 
8 


4.8 

5-6 
6.4 


566 


282 


289 


297 


305 


312 


320 


328 


335 


343 


35i 


567 


358 


366 


374 


38i 


389 


397 


404 


412 


420 


427 


9 


7. a 


568 


435 


442 


45o 


458 


465 


473 


481 


488 


496 


504 




569 

5ro 

571 


5ir 


519 


526 


534 


542 


549 


557 


565 


572 


580 




587 


595 


603 


610 


618 


626 


633 


641 


648 


656 


664 


671 


679 


686 


694 


702 


709 


717 


724 


732 


572 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 




573 


8i5 


823 


831 


838 


846 


853 


861 


868 


876 


884 




574 


891 


899 


906 


914 


921 


929 


J 37 


„ 944 


952 


959 




575 


967 


974 


982 


989 


997 


*oo5 


*OI2 


*020 


*027 


*o35 




576 


76 042 


050 


057 


065 


072 


080 


087 


095 


103 


no 




577 


118 


125 


133 


140 


148 


155 


163 


I70 


178 


185 




573 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 




579 

580 

58i 


258 


275 


283 


290 


298 


305 


313 


320 


328 


335 




343 


35o 


358 


365 


373 


380 


388 


395 


403 


410 


418 


425 


433 


440 


448 


455 


462 


470 


477 


485 


582 


492 


500 


507 


515 


522 


53o 


537 


545 


552 


559 




583 


567 


574 


582 


589 


597 


604 


612 


619 


626 


634 




584 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 




7 


585 


716 


723 


73o 


738 


745 


753 


760 


768 


775 


782 


1 
2 


0.7 

1.4 


586 


790 


797 


805 


812 


819 


827 


834 


842 


849 


856 


3 


2.1 


587 


864 


871 


879 


886 


893 


901 


908 


916 


923 


93o 


4 


2.8 


588 


938 


945 


953 


960 


967 


975 


982 


989 


997 


*oo4 


5 
6 


3-5 
4.2 


589 
590 

59i 


77 012 


019 


026 


034 


041 


048 


056 


063 


070 


078 


7 

8 

9 


4-9 
5-6 
6-3 


085 


093 


100 


107 


115 


122 


129 


137 


144 


i : 5i 


159 


166 


173 


181 


188 


195 


203 


210 


217 


225 




592 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 




593 


305 


3i3 


320 


327 


335 


342 


349 


357 


364 


371 




594 


379 


386 


393 


401 


408 


415 


422 


43o 


437 


444 




595 


452 


459 


466 


474 


481 


488 


495 


503 


5io 


517 




596 


525 


532 


539 


546 


554 


561 


568 


576 


583 


59° 




597 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 




598 


670 


677 


685 


692 


699 


706 


714 


721 


728 


735 




599 
600 


743 


75o 


757 


764 


772 


779 


786 


793 


801 


80S 




815 


822 1 830 


837 


844 


851 


859 


866 


S73 


SSo 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



13 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


600 

60 1 


77 815 


822 


830 


837 


844 


851 


859 


866 


873 


880 




887 


895 


902 


909 


916 


924 


93i 


938 


945 


952 


602 


960 


967 


974 


981 


988 


996 


*oo3 


*OIO 


*oi7 


*025 




603 


78 032 


039 


046 


053 


061 


068 


075 


082 


089 


097 




604 


104 


in 


118 


125 


132 


140 


147 


154 


161 


168 




605 


176 


183 


190 


197 


204 


211 


219 


226 


233 


240 




606 


247 


254 


262 


269 


276 


283 


290 


297 


305 


312 


s 


607 


319 


326 


333 


340 


347 


355 


362 


369 


376 


383 


t 


0.8 


608 


39° 


398 


405 


412 


419 


426 


433 


440 


447 


455 


2 


1 


t> 


609 
610 

611 


462 


469 


476 


483 


490 


497 


504 


512 


519 


526 


3 
4 

5 

6 
7 


3 
4 
4 

5 


4 
2 

8 
6 


533 


540 


547 


554 


561 


569 


576 


583 


590 


597 


604 


611 


618 


625 


633 


640 


647 


654 


661 


668 


612 


675 


682 


689 


696 


704 


711 


718 


725 


732 


739 


8 


6 


4 


613 


746 


753 


760 


767 


774 


781 


789 


796 


803 


810 


9 


7'* 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 




615 


888 


895 


902 


909 


916 


9 2 3 


* 93 ° 


J 37 


944 


951 




616 


958 


965 


972 


979 


986 


993 


*ooo 


*oo7 


*oi4 


*02I 




617 


79 029 


036 


043 


050 


057 


064 


071 


078 


085 


O92 




618 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 




619 
620 

621 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 


*T 


239 


246 


253 


260 


267 


2 74 


28l 


288 


295 


302 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 


622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


I 


0.7 


623 


449 


456 


463 


470 


477 


484 


491 


498 


505 


511 


2 


1.4 


624 


5i8 


525 


532 


539 


546 


553 


560 


567 


574 


58l 


3 


2.1 

2.8 
3.5 


625 


588 


595 


602 


609 


616 


623 


63O 


637 


644 


650 


4 

5 


626 


657 


664 


671 


678 


685 


692 


699 


706 


713 


720 


6 


4.2 


627 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 


7 
8 
9 


4-9 
5-6 

6.7 


628 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 


629 
630 

631 


865 


872 


879 


886 


893 


900 


906 


9*3 


920 


927 




934 


941 


948 


955 


962 


969 


975 


982 


989 


996 


80 003 


010 


017 


024 


030 


037 


044 


051 


058 


065 


632 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 




633 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 




634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 




635 


277 


284 


291 


298 


305 


312 


3i8 


325 


332 


339 




636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 




637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


ft 


638 


482 


489 


496 


502 


509 


516 


523 


530 


536 


543 




0.6 


639 
640 

641 


55o 


557 


564 


570 


577 


584 


59i 


598 


604 


611 


2 

3 
4 
5 
6 


1.2 

1.8 
2.4 

3-° 
3.6 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


686 


693 


699 


706 


713 


720 


726 


733 


740 


747 


642 


754 


760 


767 


774 


78i 


787 


794 


801 


808 


814 


7 


4.2 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 


8 


4.8 


644 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 


9 


5-4 


645 


956 


9 6 3 


969 


976 


983 


990 


996 


*oc>3 


*OIO 


*oi7 




646 


81 023 


030 


037 


043 


050 


057 


064 


070 


077 


084 




647 


090 


097 


104 


in 


117 


124 


131 


137 


144 


I5i 




648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 




649 
650 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 




291 


298 


305 


311 


318 


325 


33i 


338 


345 


35i 


N. 


L. | 1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. | 



LOGARITHMS. 



15 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


650 

651 


81 291 


298 


305 


311 


318 


325 


331 


338 


345 


35i 




358 


365 


37i 


378 


385 


391 


398 


405 


411 


418 


652 


425 


431 


438 


445 


45i 


458 


465 


47i 


478 


485 




653 


491 


498 


505 


5ii 


518 


525 


53i 


538 


544 


55i 




654 


558 


564 


57i 


.578 


584 


591 


598 


604 


611 


617 




655 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 




656 


690 


697 


704 


710 


717 


723 


73o 


737 


743 


75o 




657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 




658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 




659 
660 

661 


889 


895 


902 


908 


9i5 


921 


928 


935 


941 


948 


1 


954 


961 


968 


974 


981 


987 


994 


*ooo 


*oo7 


*oi4 


82 020 


027 


033 


040 


046 


053 


060 


066 


073 


079 


1 


0.7 


662 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 


2 
3 

4 


1.4 
2.1 


663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


2.8 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


5 


3-5 


665 


282 


289 


295 


302 


308 


315 


321 


328 


334 


34i 


6 
7 
8 


4-a 
4.9 
5-6 


666 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 


667 


413 


419 


426 


432 


439 


445 


452 


458 


465 


47i 


Q 


6.3 


668 


478 


484 


491 


497 


504 


5io 


517 


523 


530 


536 




669 

6ro 

671 


543 


549 


556 


562 


569 


575 


582 


588 


595 


601 




607 


614 


620 


627 


633 


640 


646 


653 


659 


666 


672 


679 


685 


692 


698 


705 


711 


718 


724 


730 


672 


737 


743 


75o 


756 


763 


769 


776 


782 


789 


795 




673 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 




674 


866 


872 


879 


885 


892 


898 


9°5 


911 


918 


924 




675 


930 


937 


943 


95o 


956 


9 6 3 


969 


975 


982 


988 




676 


995 


*OOI 


*oo8 


*oi4 


*020 


""027 


*P33 


*04o 


*o46 


*052 




677 


83 059 


065 


072 


078 


085 


091 


097 


104 


no 


117 




678 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 




679 

680 

681 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 




251 


257 


264 


270 


276 


283 


289 


296 


302 


308 


315 


321 


327 


334 


340 


347 


353 


359 


366 


372 


682 


378 


385 


39 1 


398 


404 


410 


417 


423 


429 


436 




683 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 




684 


506 


512 


5i8 


525 


531 


537 


544 


55o 


556 


563 




0.6 
1.2 


685 


569 


575 


582 


588 


594 


601 


607 


613 


620 


626 


2 


686 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 


3 


1.8 


687 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 


4 


2.4 


688 


759 


765 


771 


778 


784 


790 


797 


803 


809 


816 


5 
6 


3-° 
3-6 


689 
690 

691 


822 


828 


835 


841 


847 


853 


860 


866 


872 


879 


7 
8 

9 


4.2 

4 .8 
5-4 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 


948 


954 


960 


967 


973 


979 


985 


992 


998 


*oo4 




692 


84 on 


017 


023 


029 


036 


042 


048 


055 


061 


067 




693 


073 


080 


086 


092 


098 


105 


in 


117 


123 


130 




694 


136 


142 


148 


155 


161 


167 


173 


180 


186 


192 




695 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 




696 


261 


267 


273 


280 


286 


292 


298 


305 


3ii 


317 




697 


323 


330 


336 


342 


348 


354 


361 


367 


373 


379 




698 


386 


39 2 


398 


404 


410 


4i7 


423 


429 


435 


442 




699 

roo 


448 


454 


460 


466 


473 


479 


485 


491 


497 


504 

566 




510 


5i6 


522 


528 


535 


54i 


547 


553 


559 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



u 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


700 

701 


84 510 


516 


522 


528 


535 


54i 


547 


553 


559 


566 




572 


578 


584 


59° 


597 


603 


609 


615 


621 


628 


702 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 




703 


696 


702 


708 


714 


720 


726 


733 


739 


745 


75i 




704 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 




705 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 




706 


880 


887 


893 


899 


9° 5 


911 


917 


924 


930 


936 




707 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 


7 


708 


85 003 


009 


016 


022 


028 


034 


040 


046 


052 


058 


1 
2 


0.7 
1 .4 


709 

no 

711 


065 


071 


077 


083 


089 


095 


101 


107 


114 


120 


3 
4 
5 

6 


2.1 
2.8 
3-5 
4.2 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 


712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 


7 
8 


4-9 
5-6 


713 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 


9 


6-3 


714 


370 


376 


382 


388 


394 


400 


406 


412 


418 


425 




715 


43i 


437 


443 


449 


455 


461 


467 


473 


479 


485 




716 


491 


497 


503 


509 


516 


522 


528 


534 


540 


546 




717 


552 


558 


564 


57o 


576 


582 


588 


594 


600 


606 




718 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 




719 

T20 

721 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 




733 


739 


745 


751 


757 


763 


769 


775 


78i 


788 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 


722 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 




O 

0.6 


723 


914 


920 


926 


932 


938 


944 


95o 


956 


962 


968 


2 


1.2 


724 


974 


980 


986 


992 


998 


*oo4 


*OIO 


*oi6 


*022 


*028 


3 


1.8 


725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 


4 
5 
6 


2.4 


726 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 


3-6 


727 


153 


159 


165 


171 


177 


183 


189 


195 


20I 


207 


7 


4.2 


728 


213 


219 


225 


231 


237 


243 


249 


255 


26l 


267 


8 

Q 


4.8 

e 4 


729 
730 

73i 


273 

332 


279 

338 


285 


291 


297 


303 


308 


314 


320 


326 




344 


350 


356 


362 


368 


374 


38o 


386 


392 


393 


404 


410 


415 


421 


427 


433 


439 


445 


732 


45i 


457 


463 


469 


475 


481 


487 


493 


499 


504 




733 


5io 


5i6 


522 


528 


534 


540 


546 


552 


558 


564 




734 


57o 


576 


58i 


587 


593 


599 


605 


611 


617 


623 




735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 




73b 


688 


694 


700 


705 


7ii 


717 


723 


729 


735 


741 




737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 


5 


738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


1 


o-5 


739 
740 

74i 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


3 
4 

5 
6 


i-5 

2.0 

2-5 

3.0 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 


982 


988 


994 


999 


*oo5 


*OII 


*oi7 


*023 


*029 


*035 


742 


87 040 


046 


052 


058 


064 


070 


075 


081 


087 


093 


7 

8 


3-5 
4.0 
4-5 


743 


099 


105 


in 


116 


122 


128 


134 


140 


146 


151 


9 


744 


157 


163 


169 


175 


181 


1S6 


192 


198 


204 


210 




745 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 




746 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 




747 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 




743 


390 


39 6 


402 


408 


413 


419 


425 


43i 


437 


442 




749 
750 


448 


454 


460 


466 


47i 


477 


483 


489 


495 


500 




506 


512 


5i8 


523 


529 


535 


54i 


547 


552 


558 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



15 



N. 


L. 


i 


2 


3 


4 


5 


6 


7 


8 


9 


P. P 


750 

75i 


87 506 


512 


518 


523 


529 


535 


54i 


547 


552 


558 




564 


570 


576 


581 


587 


593 


599 


604 


610 


616 


752 


622 


628 


633 


639 


645 


651 


656 


662 


668 


674 




753 


679 


685 


69I 


697 


703 


708 


714 


720 


726 


73i 




754 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




755 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 




756 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




757 


910 


9i5 


921 


927 


933 


938 


944 


95o 


955 


961 




758 


967 


973 


978 


984 


990 


996 


*OOI 


*oo7 


*oi3 


*oi8 




759 
760 

761 


88 024 


030 


036 


041 


047 


053 


058 


064 


070 


076 


A. 


081 


087 


093 


098 


104 


no 


116 


121 


127 


133 


138 


144 


150 


156 


161 


167 


173 


178 


184 


190 


762 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 


I 


0.6 


763 


252 


258 


264 


270 


275 


281 


287 


292 


298 


304 


2 


1.2 


764 


309 


315 


321 


326 


332 


338 


343 


349 


355 


360 


3 


1.8 


765 


366 


372 


377 


383 


389 


395 


400 


406 


412 


417 


4 

5 


2.4 
3-° 


766 


423 


429 


434 


440 


446 


45i 


457 


463 


468 


474 


6 


3-6 


767 


480 


485 


491 


497 


502 


508 


513 


519 


525 


530 


7 


4.2 


768 


536 


542 


547 


553 


559 


564 


57o 


576 


'581 


587 


8 
g 


4.8 

C .A 


769 

rro 

771 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 




649 


655 


660 


666 


672 


677 


683 


689 


694 


700 


7o5 


711 


717 


722 


728 


734 


739 


745 


75o 


756 


772 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 




773 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 




774 


874 


880 


885 


891 


897 


902 


908 


913 


919 


925 




775 


930 


936 


941 


* 947 


953 


958 


J> 64 


969 


975 


981 




776 


986 


992 


997 


*oo3 


*oo9 


*oi4 


*020 


*025 


*03i 


*o37 




777 


89 042 


048 


o53 


059 


064 


070 


076 


081 


087 


092 




778 


098 


104 


109 


ii5 


120 


126 


131 


137 


143 


148 




779 

780 

781 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 


5 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 


265 


271 


276 


282 


287 


293 


298 


304 


310 


315 


782 


321 


326 


332 


337 


343 


348 


354 


360 


365 


37i 


1 


0.5 


786 


376 


382 


387 


393 


398 


404 


409 


415 


421 


426 


2 
3 
4 


I .5 


784 


432 


437 


443 


448 


454 


459 


465 


470 


476 


481 


2.0 


785 


487 


492 


498 


504 


509 


515 


520 


526 


531 


537 


5 


2-5 


786 


542 


548 


553 


559 


564 


57o 


575 


581 


586 


592 


6 


3-o 
3-5 
4.0 


787 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 


7 
8 


788 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 


9 


4-5 


789 
790 

791 


708 


713 


719 


724 


730 


735 


741 


746 


752 


757 




763 


768 


774 


779 


785 


790 


796 


801 


807 


812 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 


792 


873 


878 


883' 


889 


894 


900 


9° 5 


911 


916 


922 




793 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




794 


982 


988 


993 


998 


*oo4 


*oo9 


*oi5 


*020 


*026 


*c>3 1 




795 


90 037 


042 


048 


053 


059 


064 


069 


075 


080 


086 




796 


091 


097 


102 


108 


113 


119 


124 


129 


135 


140 




797 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 




798 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 




799 
800 


255 


260 


266 


271 


276 


282 


287 


293 


298 


304 




309 


3i4 


320 


325 


33i 


336 


342 


347 


352 


35? 


N. 


L. 


1 


2 


3 


4 


5. 


6 


7 


8 


9 


P. P. 



16 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


800 

801 


90 309 


314 


320 


325 


33i 


336 


342 


347 


352 


358 




363 


369 


374 


380 


385 


39° 


396 


401 


407 


412 


802 


417 


423 


428 


434 


439 


445 


45o 


455 


461 


466 




803 


472 


477 


482 


488 


493 


499 


504 


509 


515 


520 




804 


526 


53i 


536 


542 


547 


553 


558 


563 


569 


574 




805 


580 


585 


59° 


596 


601 


607 


612 


617 


623 


628 




806 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 




807 


687 


693 


698 


703 


709 


714 


720 


725 


73o 


736 




808 


741 


747 


752 


757 


763 


768 


773 


779 


.784 


789 




809 
810 

811 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 




849 


854 


859 


865 


870 


875 


881 


886 


891 


897 


902 


907 


9 J 3 


918 


924 


929 


934 


940 


945 


95o 


812 


956 


961 


966 


972 


977 


982 


988 


993 


998 


*oo4 


6 


813 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


o57 


1 








814 


062 


068 


073 


078 


084 


089 


094 


100 


105 


no 


3 


j 


8 


815 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 


4 


2 


4 


816 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 


5 


3 



6 


817 


222 


228 


233 


238 


243 


249 


254 


259 


265 


270 


7 


3 
4 


2 


818 


275 


281 


286 


291 


297 


302 


307 


312 


3i8 


323 


8 


4 


8 


819 
820 

821 


328 


334 


339 


344 


35o 


355 


360 


365 


37i 


376 


9 


5-4 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 




434 


440 


445 


45o 


455 


461 


466 


471 


477 


482 


822 


487 


492 


498 


503 


508 


514 


519 


524 


529 


535 




823 


540 


545 


55i 


556 


56i 


566 


572 


577 


582 


587 




824 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 




825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


693 




826 


698 


703 


709 


714 


719 


724 


730 


735 


740 


745 




827 


751 


756 


761 


766 


772 


777 


782 


787 


793 


798 




828 


803 


808 


814 


819 


824 


829 


834 


840 


845 


850 




829 
830 

831 


855 


861 


866 


871 


876 


882 


887 


892 


897 


903 


5 


908 


913 


918 


924 


929 


934 


939 


944 


95o 


955 


960 


9 6 5 


971 


976 


981 


986 


991 


997 


*002 


*oo7 


832 


92 012 


018 


023 


028 


033 


038 


044 


049 


054 


059 


1 


o-S 


833 


065 


070 


o75 


080 


085 


091 


096 


IOI 


T06 


in 


3 


r -5 


834 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 


4 


2.0 


835 


169 


174 


179 


184 


189 


195 


200 


205 


2IO 


215 


5 


2-5 


836 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 


7 


3-° 
3-5 


837 


273 


278 


283 


288 


293 


298 


304 


309 


314 


319 


8 


4.0 


838 


324 


330 


335 


34o 


345 


35o 


355 


361 


366 


37i 


9 


4-5 


839 
840 

841 


376 


381 


387 


392 


397 


402 


407 


412 


418 


423 




428 


433 


438 


443 


449 


454 


459 


464 


469 


474 


480 


485 


490 


495 


500 


505 


5ii 


5i6 


521 


526 


842 


531 


536 


542 


547 


552 


557 


562 


567 


572 


578 




843 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 




844 


634 


639 


645 


650 


655 


666 


665 


670 


675 


681 




845 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 




846 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 




847 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 




848 


840 


845 


850 


855 


860 


865 


870 


875 


88l 


886 




849 
850 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 




942 


947 


952 


957 


962 


967 


973 


978 


983 


988 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



17 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


850 

851 


92 942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




993 


998 


*oo3 


*oo8 


*oi3 


*oi8 


*024 


*029 


*034 


*o39 


852 


93 044 


049 


054 


059 


064 


069 


075 


080 


085 


090 




853 


095 


100 


105 


no 


ii5 


120 


125 


131 


136 


141 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




855 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 




856 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 




857 


298 


303 


308 


313 


3i8 


323 


328 


334 


339 


344 


6 


858 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


1 
2 






2 


859 
860 

861 


399 


404 


409 


414 


420 


425 


43o 


435 


440 


445 


3 
4 
5 
6 


2 

3 
3 


8 
4 

6 


45o 


455 


460 


465 


470 


475 


480 


485 


490 


495 


500 


505 


5io 


515 


520 


526 


53i 


536 


54i 


546 


862 


551 


556 


56i 


566 


571 


576 


581 


586 


59i 


596 


7 
8 


4 
4 


2 
8 


863 


601 


606 


611 


6l6 


621 


626 


631 


636 


641 


646 


9 


5 


4 


864 


651 


656 


661 


666 


671 


676 


682 


687 


692 


697 




865 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




869 

sro 

871 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




952 


957 


962 


967 


972 


977 


982 


987 


992 


997 


94 002 


007 


012 


017 


022 


027 


032 


037 


042 


047 


872 


052 


057 


062 


067 


072 


077 


082 


086 


091 


096 




9 


873 


IOI 


106 


in 


116 


121 


126 


131 


136 


141 


146 


a 


0.5 
1.0 


874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 


3 


i-5 


875 


201 


206 


211 


216 


221 


226 


231 


236 


240 


245 


4 


2.0 


876 


250 


255 


260 


265 


270 


275 


280 


285 


290 


295 


5 
6 


2-5 

3-o 


877 


300 


305 


310 


315 


320 


325 


33o 


335 


34o 


345 


7 


3-5 


878 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


8 



4.0 

a - e 


879 

880 

881 


399 


404 
453 


409 


414 


419 


424 


429 


433 


438 


443 


y t-w 


448 


458 


463 


468 


473 


478 


483 


488 


493 


498 


503 


507 


512 


517 


522 


527 


532 


537 


542 


882 


547 


552 


557 


562 


567 


571 


576 


581 


586 


591 




883 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 




884 


645 


650 


655 


660 


665 


670 


675 


680 


685 


689 




885 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 




886 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 




887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 


A 


888 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 


1 





4 

g 


889 

890 

891 


890 


895 


900 


9°5 


910 


915 


919 


924 


929 


934 


3 
4 

5 
6 


1 
1 
2 

2 


2 

6 

4 


939 


944 


949 


954 


959 


9 6 3 


968 


973 


978 


983 


988 


993 


998 


*002 


*oo7 


*OI2 


*oi7 


*022 


*02 7 


*032 


892 


95 036 


041 


046 


05I 


056 


o6l 


066 


07I 


075 


080 


7 
8 


2 


8 


893 


085 


090 


095 


IOO 


105 


IO9 


114 


119 


124 


129 


9 


3 
3 


6 


894 


134 


139 


143 


I48 


153 


158 


163 


168 


173 


177 




895 


182 


187 


192 


197 


202 


207 


211 


2l6 


221 


226 




896 


231 


236 


240 


245 


250 


255 


260 


265 


270 


274 




897 


279 


284 


289 


294 


299 


303 


308 


313 


318 


323 




898 


328 


332 


337 


342 


347 


352 


357 


36l 


366 


371 




899 
900 


376 
424 


381 


386 


390 


395 


4OO 


405 


4IO 


415 


419 




429 


434 


439 


444 


448 


453 


458 


463 


468 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



18 



LOGARITHMS. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


900 

901 


95 424 


429 


434 


439 


444 


448 


453 


458 


463 


468 




472 


477 


482 


487 


492 


497 


501 


506 


511 


516 


902 


521 


525 


530 


535 


540 


545 


550 


554 


559 


564 




903 


569 


574 


578 


583 


588 


593 


598 


602 


607 


612 




904 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 




9°5 


665 


670 


674 


679 


684 


689 


694 


698 


703 


708 




906 


713 


718 


722 


727 


732 


737 


742 


746 


75i 


756 




907 


761 


766 


770 


775 


7S0 


785 


789 


794 


799 


804 




90S 


809 


813 


818 


823 


828 


832 


837 


842 


847 


852 




909 
910 

911 


856 


861 


866 


871 


875 


880 


885 


890 


895 


899 


S 


904 


909 


914 


918 


9 2 3 


928 


933 


938 


942 


947 


952 


957 


961 


966 


971 


976 


9S0 


985 


*"° 


995 


z 


0.5 


912 


999 


^004 


*oo9 


*oi4 


*oi9 


*C>23 


*028 


*033 


*o 3 8 


*042 


2 


1 .0 
2.0 


9 J 3 


96 047 


052 


o57 


061 


066 


071 


076 


080 


085 


090 


3 
4 


914 


095 


099 


104 


109 


114 


118 


123 


128 


133 


137 


5 


2-5 


915 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 


6 
7 
8 


3-o 
3 5 
+ .0 


916 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 


917 


237 


242 


246 


251 


256 


261 


265 


270 


275 


280 


9 


4 5 


918 


284 


289 


294 


298 


303 


308 


313 


317 


322 


327 




919 
920 

921 


332 


336 


34i 


346 


350 


355 


360 


365 


369 


374 




379 


384 


388 


393 


398 


402 


407 


412 


417 


421 


426 


43i 


435 


440 


445 


45o 


454 


459 


464 


468 


922 


473 


478 


483 


487 


492 


497 


501 


506 


5ii 


515 




9 2 3 


520 


525 


530 


534 


539 


544 


548 


553 


558 


562 




924 


567 


572 


577 


581 


586 


591 


595 


600 


605 


609 




9 2 5 


614 


619 


624 


628 


633 


638 


642 


647 


652 


656 




926 


661 


666 


670 


675 


680 


685 


689 


694 


699 


703 




927 


708 


713 


717 


722 


727 


73i 


736 


741 


745 


75o 




928 


755 


759 


764 


769 


774 


778 


783 


788 


792 


797 




929 
930 

931 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 




848 


853 


858 


862 


867 


872 


876 


881 


886 


890 


895 


900 


904 


909 


914 


918 


923 


928 


932 


937 


932 


942 


946 


951 


956 


* 96 ° 


965 


970 


974 


979 


984 




933 


988 


993 


997 


"002 


*oo7 


*OII 


*oi6 


*02I 


*02 5 


^030 




934 


97 035 


039 


044 


049 


053 


058 


063 


067 


072 


077 




4 


935 


081 


086 


090 


095 


100 


104 


109 


114 


118 


123 


2 





8 


936 


128 


132 


137 


142 


146 


151 


155 


l6o 


165 


169 


3 


1 


2 


937 


174 


-79 


183 


188 


192 


197 


202 


206 


211 


216 


4 


1 


6 


938 


220 


225 


230 


234 


239 


243 


248 


253 


257 


262 


5 
6 


2 


4 


939 
940 

941 


267 


271 


276 


280 


285 


290 


294 


299 


304 


308 


7 

8 
9 


2 


8 


313 


317 


322 


327 


33i 


336 


340 


345 


350 


354 


3 l 
3-6 


359 


364 


368 


373 


377 


382 


387 


39 1 


39 6 


400 




942 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




943 


451 


456 


460 


465 


470 


474 


479 


483 


4S8 


493 




944 


497 


502 


506 


5ii 


516 


520 


525 


529 


534 


539 




945 


543 


548 


552 


557 


562 


566 


57i 


575 


580 


585 




946 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 




947 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 




948 


681 


685 


690 


695 


699 


704 


708 


713 


717 


722 




949 
950 


727 


731 


736 


740 


745 


749 


754 


759 


763 


768 




772 


777 


782 


7S6 


70 T 


795 


800 


804 


809 


813 


In. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



LOGARITHMS. 



19 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


950 

95i 


97 772 


777 


782 


786 


791 


795 


800 


804 


809 


813 




818 


823 


827 


832 


836 


841 


845 


850 


855 


859 


952 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 




953 


909 


914 


918 


9 2 3 


928 


932 


937 


941 


946 


95o 




954 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 




955 


98 000 


005 


009 


014 


019 


023 


028 


032 


037 


041 




956 


046 


050 


o55 


059 


064 


068 


073 


078 


082 


087 




957 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 




958 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 




959 
960 

961 


182 


186 


191 


i95 


200 


204 


209 


214 


218 


223 




227 


232 


236 


241 


245 


250 


~254 


259 


263 


268 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 


962 


' 3i8 


322 


327 


33i 


336 


340 


345 


349 


354 


3SB 


1 


5 

1.0 


9 6 3 


363 


367 


372 


376 


381 


385 


39° 


394 


399 


403 


2 


964 


408 


412 


4i7 


421 


426 


430 


435 


439 


444 


448 


3 


i-5 


9 6 5 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 


4 

5 


2.0 

2 -5 


966 


498 


502 


507 


511 


5i6 


520 


525 


529 


534 


538 


6 


3-o 


967 


543 


547 


552 


556 


56i 


565 


57o 


574 


579 


583 


" 


3-5 


968 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


S 

Q 


4.0 

A C 


969 
970 

97i 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 




677 


682 


686 


691 


695 


700 


704 


709 


713 


717 


722 


726 


73i 


735 


740 


744 


749 


753 


758 


762 


972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 




973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 




974 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 




975 


900 


9°5 


909 


914 


918 


923 


927 


932 


936 


941 




976 


945 


949 


954 


958 


9 6 3 


967 


972 


976 


981 


985 




977 


989 


994 


998 


*oo3 


*oc>7 


*OI2 


*oi6 


*02I 


*025 


*02g 




978 


99 °34 


038 


043 


047 


052 


O56 


061 


065 


069 


074 




979 
980 

981 


078 


083 


087 


092 


096 


IOO 


105 


IO9 


114 


118 


4 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 


167 


171 


176 


180 


185 


189 


193 


I98 


202 


207 


982 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 


i 


O.4 


983 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 


3 


O.8 

1.2 


984 


300 


304 


308 


313 


317 


322 


326 


330 


335 


339 


4 


1.6 


985 


344 


348 


352 


357 


361 


366 


3 70 


374 


379 


383 


5 


2.0 


986 


388 


392 


39 6 


401 


405 


4IO 


414 


419 


423 


427 


6 
7 


2.4 
2.8 


987 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 


8 


3-2 


988 


476 


480 


484 


489 


493 


498 


502 


506 


5ii 


515 


9 


3.6 


989 
990 

991 


520 


524 


528 


533 


537 


542 


546 


55o 


555 


559 




564 


568 


572 


577 


58i 


585 


59° 


594 


599 


603 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 


992 


651 


656 


660 


664 


669 


673 


677 


6S2 


686 


691 




993 


695 


699 


704 


708 


712 


717 


721 


726 


730 


734 




994 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 




995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 




996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 




997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 




998 


9*3 


917 


922 


926 


930 


935 


939 


944 


94S 


952 




999 
1000 


957 


961 


9 6 5 


970 


974 


978 


983 


987 


991 


996 




00 000 


004 


009 


013 


017 


022 


026 


030 


035 


039 


N. 


L. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 



TABLES 



OF 

NATURAL SINES, COSINES 
TANGENTS, 
AND COTANGENTS 

GIVING THE VALUES OF THE FUNCTIONS FOR 
ALL DEGREES AND MINUTES FROM 
0° TO 90° 



Natural Sines and Cosines. 



23 



f 








1 '• 


2 





3° 


4° 


t 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


'osine 


Sine C 


Cosine 


o 


.00000 




.01745 


.99985 


.03490 


•99939 


•05234 


99863 


.06976 


99756 


60 


I 


.00029 






.01774 


.99984 


.03519 


.99938 


.05263 


99861 


.07005 


99754 


59 


2 


.00058 






.01803 


.99984 


•03548 


•99937 


.05292 


99S60 


•07034 


99752 


58 


3 


.00087 






.01832 


.99983 


•03577 


.99936 


.05321 


99858 


.07063 


99750 


57 


4 


.00116 






.01862 


.99983 


.03606 


•99935 


.05350 


99857 


.07092 


99748 


56 


5 


.00145 






.01891 


.99982 


•03635 


•99934 


•05379 


99855 


.07121 


99746 


55 


6 


.00175 






.01920 


.99982 


.03664 


•99933 


.05408 


99854 


.07150 


99744 


54 


7 


.00204 






.01949 


.99981 


.03693 


.99932 


.05437 


99852 


.07179 


99742 


53 


S 


.00233 






.01978 


.99980 


•03723 


.99931 


.05466 


99851 


.07208 


99740 


52 


9 


.00262 






.02007 


.99980 


•03752 


.99930 


.05495 


99849 


•07237 


99738 


5i 


IO 


.00291 






.02036 


.99979 


.03781 


.99929 


.05524 


99847 


.07266 


99736 


50 


ii 


.00320 


.99999 


.02065 


.99979 


.03810 


.99927 


•05553 


99846 


•07295 


99734 


49 


12 


.00349 


.99999 


.02094 


•99978 


.03839 


.99926 


.05582 


99844 


.07324 


9973 1 


48 


13 


.00378 


.99999 


.02123 


.99977 


.03868 


.99925 


.05611 


99842 


.07353 


99729 


47 


14 


. 00407 


.99999 


.02152 


.99977 


.03897 


.99924 


.05640 


99841 


.07382 


99727 


46 


15 


.00436 


.99999 


.02181 


.99976 


.03926 


.99923 


.05669 


99839 


.07411 


99725 


45 


16 


.00465 


.99999 


.02211 


.99976 


•03955 


.99922 


.05698 


99838 


.07440 


99723 


44 


17 


.00495 


.99999 


.03240 


•99975 


.03984 


.99921 


•05727 


99836 


.07469 


99721 


43 


18 


.00524 


.99999 


.02269 


.99974 


.04013 


.99919 


•05756 


99834 


.07498 


99719 


42 


19 


•00553 


.99998 


.02298 


.99974 


.04042 


.99918 


•05785 


99833 


.07527 


99716 


4i 


20 


.00582 


.99998 


.02327 


•99973 


.04071 


•99517 


.05814 


99831 


•07556 


997*4 


40 


21 


.00611 


.99998 


.02356 


.99972 


.04100 


.99916 


.05844 


99829 


.07585 


99712 


39 


22 


.00640 


•99998 


.02385 


.99972 


.04129 


.99915 


.05873 


99827 


.07614 


99710 


38 


23 


.00669 


•99998 


.02414 


.99971 


.04159 


•99913 


.05902 


99826 


•07643 


99708 


37 


24 


.00698 


•99993 


.02443 


.99970 


.04188 


.99912 


•05931 


99824 


.07672 


99705 


36 


25 


.00727 


.99997 


.02472 


.99969 


.04217 


.99911 


.05960 


99822 


.07701 


99703 


35 


26 


.00756 


.99997 


.02501 


.99969 


.04246 


.99910 


.05989 


99821 


•07730 


99701 


34 


27 


.00785 


.99997 


.02530 


.99968 


•04275 


.99909 


.06018 


99819 


•07759 


99699 


33 


28 


.00814 


.99997 


.02560 


.99967 


.04304 


.99907 


.06047 


99817 


.07788 


99696 


32 


29 


.00844 


.99996 


.02589 


.99966 


•04333 


.99906 


.06076 


99815 


.07817 


99694 


3i 


30 


.00873 


•99996 


.02618 


.99966 


.04362 


.99905 


.06105 


99813 


.07846 


99692 


30 


3 1 


00902 


.99996 


.02647 


.99965 


.04391 


.99904 


.06134 


99812 


.07875 


99689 


29 


32 


,00931 


.99996 


.02676 


.99964 


.04420 


.99902 


.06163 


99810 


.07904 


99687 


28 


33 


.00960 


•99995 


.02705 


.99963 


.04449 


.99901 


.06192 


99808 


•07933 


99685 


27 


34 


.00989 


•99995 


•02734 


.99963 


.04478 


. 99900 


.06221 


99806 


.07962 


99683 


26 


35 


.01018 


•99995 


.02763 


.99962 


.04507 


.99898 


.06250 


99804 


.07991 


99680 


25 


36 


.01047 


•99995 


.02792 


.99961 


.04536 


.99897 


.06279 


99803 


.08020 


99678 


24 


37 


.01076 


.99994 


.02821 


.99960 


.04565 


.99896 


.06308 


99801 


.08049 


99676 


23 


38 


.01105 


.99994 


.02850 


•99959 


.04594 


.99894 


•06337 


99799 


.08078 


99673 


22 


39 


.01134 


•99994 


.02879 


•99959 


.04623 


.99893 


.06366 


99797 


.08107 


99671 


21 


4° 


.01164 


•99993 


.02908 


.99958 


•04653 


.99892 


•06395 


99795 


.08136 


99668 


20 


4i 


.01193 


•99993 


.02938 


•99957 


.04682 


.99890 


.06424 


99793 


.08165 


99666 


19 


42 


.01222 


•99993 


.02967 


.99950 


.04711 


.99889 


.06453 


99792 


.08194 


99664 


18 


43 


.01251 


•99992 


.02996 


•99955 


.04740 


.99888 


.06482 


99790 


.08223 


99661 


17 


44 


.01280 


•Q9992 


.03025 


•99954 


.04769 


.99886 


.06511 


99788 


.08252 


99659 


16 


45 


.01309 


.99991 


.03054 


•99953 


.04798 


.99885 


.06540 


99786 


.08281 


99657 


15 


46 


.01338 


.99991 


.03083 


.99952 


.04827 


.99883 


.06569 


99784 


.08310 


99654 


14 


47 


.01367 


.99991 


.03112 


•99952 


.04856 


.99882 


.06598 


99782 


.08339 


99652 


J 3 


48 


.01396 


.99990 


.03141 


•99951 


.04885 


.99881 


.06627 


99780 


.0S368 


99649 


12 


49 


.01425 


•99990 


.03170 


.99950 


.04914 


.99879 


.06656 


99778 


.08397 


99647 


11 


S° 


.01454 


.99989 


.03199 


.99949 


.04943 


.99878 


.06685 


99776 


.08426 


99644 


10 


5i 


.01483 


.99989 


.03228 


.99948 


.04972 


.99876 


.06714 


99774 


.08455 


99642 


9 


52 


•01513 


.99989 


.03257 


.99947 


.05001 


•99875 


•06743 


99772 


.08484 


99639 


8 


53 


.01542 


.99988 


.03286 


.99946 


.05030 


•99873 


.06773 


99770 


.08513 


99637 


7 


54 


.01571 


.99988 


.03316 


•99945 


•05059 


.99872 


.06802 


99768 


.08542 


99635 


6 


55 


.01600 


.99987 


•03345 


.99944 


.05088 


.99870 


.06831 


99766 


.08571 


99632 


5 


56 


.01629 


.99987 


•03374 


•99943 


.05117 


.99869 


.06860 


99764 


.0S600 


99630 


4 


57 


.01658 


.99986 


•03403 


.99942 


.05146 


.99867 


.06889 


99762 


.0S629 


99627 


3 


58 


.01687 


.99986 


.03432 


.99941 


•05175 


.99866 


. 069 1 8 


99760 


.0S658 


99625 


2 


59 


.01716 


.99985 


.03461 


.99940 


.05205 


.99S64 


.06947 


99758 


.0S687 


99622 


1 


60 


•01745 


.99985 


.03490 


•99939 


•05234 


.99863 


.06976 


00756 


.08716 


00610 





f 
— 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


t 


8 


f 


8B 





8 


7° 


86 


2 


35 


> 



a. G. IV.— 2k 



24 



Natural Sines and Cosines. 



1 


5° 


6° 


7° 


8° 


9 





1 


Sine C 


osine 


Sine C 


osine 


Sine C 


losine 


Sine C 


'osine 


Sine 


Cosine 


o 


.08716 


99619 


•10453 


99452 


.12187 


99255 


.13917 


99027 


.15643 


.98769 


60 


I 


.08745 


99617 


. 10482 


99449 


.12216 


99251 


.13946 


99023 


.15672 


.98764 


59 


2 


.08774 


996:4 


.10511 


99446 


.12245 


99248 


.13975 


99019 


.15701 


.9S760 


58 


3 


.08803 


99612 


. 10540 


99443 


.12274 


99244 


. 14004 


99015 


•15730 


-98755 


57 


4 


.08831 


99609 


. 10569 


99440 


.12302 


99240 


.14033 


990 1 1 


•15758 


•98751 


56 


5 


.08860 


99607 


• 10597 


99437 


.12331 


99237 


.14061 


99006 


•15787 


.98746 


55 


6 


.08889 


99604 


.10626 


99434 


.12360 


99233 


.14090 


99002 


.15816 


.98741 


54 


7 


.08918 


99602 


.10655 


99431 


.12389 


99230 


.14119 


98998 


.15845 


•98737 


53 


8 


.08947 


99599 


. 10684 


99428 


.12418 


99226 


.14148 


98994 


.15873 


•98732 


52 


9 


.08976 


99596 


.10713 


99424 


.12447 


99222 


.14177 


98990 


.15902 


.98728 


5i 


IO 


.09005 


99594 


. 10742 


99421 


.12476 


99219 


.14205 


98986 


.15931 


.98723 


50 


:i 


•09034 


99591 


.10771 


99418 


. 12504 


99215 


.14234 


98982 


•15959 


.98718 


49 


12 


.09063 


99588 


.10800 


99415 


•12533 


992 1 1 


.14263 


98978 


.15988 


.98714 


48 


13 


.09092 


99586 


.10829 


99412 


.12562 


99208 


.14292 


98973 


.16017 


.98709 


47 


14 


.09121 


99583 


.10858 


99409 


.12591 


99204 


.14320 


98969 


.16046 


.98704 


46 


15 


.09150 


99580 


.10887 


99406 


.12620 


99200 


•14349 


98965 


.16074 


.98700 


45 


16 


.09179 


99578 


.10916 


99402 


.12649 


99197 


.14378 


98961 


.16103 


.98695 


44 


17 


.09208 


99575 


• 10945 


99399 


.12678 


99193 


• i44°7 


98957 


.16132 


.08690 


43 


18 


.09237 


99572 


.10973 


99396 


.12706 


99189 


• 14436 


98953 


.16160 


.98686 


42 


19 


.09266 


99570 


.11002 


99393 


• I 2735 


99186 


. 14464 


98948 


.16189 


.98681 


41 


20 


.09295 


99567 


.11031 


9939° 


.12764 


99182 


•14493 


98944 


.16218 


.98676 


40 


21 


.09324 


99564 


.11060 


99386 


•12793 


99178 


.14522 


98940 


.16246 


.98671 


39 


22 


•09353 


99562 


.11089 


99383 


.12822 


99175 


.14551 


98936 


.16275 


.98667 


38 


23 


.09382 


99559 


.11118 


90380 


.12851 


99171 


.14580 


98931 


.16304 


.98662 


37 


24 


.09411 


99556 


.11147 


99377 


.12880 


99167 


.14608 


98927 


.16333 


.98657 


36 


25 


.09440 


99553 


.11176 


99374 


.12908 


99163 


.14637 


98923 


.1636! 


.98652 


35 


26 


.09469 


99551 


.11205 


99370 


•12937 


99160 


.14666 


98919 


• 16390 


.98648 


34 


27 


.09498 


99548 


.11234 


99367 


.12966 


99156 


.14695 


98914 


.16419 


.98643 


33 


28 


.09527 


99545 


.11263 


99364 


.12995 


99152 


•14723 


98910 


•16447 


.98638 


32 


29 


•09556 


99542 


.11291 


99360 


.13024 


99M8 


.14752 


98906 


.16476 


.98633 


3i 


3° 


.09585 


99540 


.11320 


99357 


•13053 


99144 


.14781 


98902 


• 16505 


. 98629 


30 


3i 


.09614 


99537 


•1 1349 


99354 


.13081 


99141 


.14810 


98897 


•16533 


.98624 


29 


32 


.09642 


99534 


.11378 


99351 


.13110 


99137 


.14838 


98893 


.16562 


.98619 


28 


33 


.09671 


99531 


.11407 


99347 


•13139 


99133 


.14867 


98889 


.16591 


.98614 


27 


34 


.09700 


99528 


.11436 


99344 


.13168 


99129 


.14896 


98884 


.16620 


.98609 


26 


35 


.09729 


99526 


.11465 


99341 


• I 3 I 97 


99 J 25 


•14925 


98880 


.16648 


.98604 


25 


36 


.09758 


99523 


.11494 


99337 


.13226 


99122 


•14954 


98876 


.16677 


.98600 


24 


37 


.09787 


99520 


.11523 


99334 


•13254 


99118 


.14982 


98871 


.16706 


•98595 


23 


38 


.09816 


99517 


•11552 


99331 


.13283 


991 14 


.15011 


98867 


.16734 


.98590 


22 


39 


.09845 


99514 


.11580 


99327 


.13312 


99110 


.15040 


98863 


.16763 


.98585 


21 


40 


.09874 


995i 1 


.11609 


993 2 4 


•I334I 


99106 


.15069 


98858 


.16792 


.98580 


20 


41 


.09903 


99508 


.11638 


90320 


•13370 


99102 


•15097 


98854 


.16820 


•98575 


19 


42 


.09932 


99506 


.11667 


993 x 7 


•13399 


99098 


.15126 


98849 


.16849 


.98570 


18 


43 


.09961 


99503 


.11696 


99314 


•13427 


99094 


.15155 


98845 


.16878 


•98565 


17 


44 


.09990 


99500 


.11725 


99310 


•13456 


99091 


.15184 


98841 


.16906 


.98561 


16 


45 


.10019 


99497 


.11754 


99307 


.13485 


99087 


.15212 


98836 


.16935 


.98556 


15 


46 


.10048 


99494 


.11783 


99303 


•I35H 


99083 


.15241 


98832 


.16964 


•98551 


14 


47 


.10077 


99491 


.11812 


99300 


•13543 


99079 


.15270 


98827 


.16992 


.98546 


13 


48 


.10106 


99488 


.11840 


09297 


•13572 


99075 


•15299 


98823 


.17021 


.98541 


12 


49 


.10135 


99485 


.11869 


99293 


.13600 


99071 


•15327 


98S18 


.17050 


.98536 


11 


50 


.10164 


99482 


.11898 


99290 


.13629 


99067 


•15356 


98814 


.17078 


.98531 


10 


5i 


.10192 


99479 


.11927 


99286 


.13658 


99063 


.15385 


98809 


.17107 


.98526 


1 


52 


.10221 


99476 


.11956 


99283 


.13687 


99059 


.15414 


98805 


.17136 


.98521 


53 


. 10250 


99473 


.11985 


99279 


.13716 


99°55 


.15442 


98800 


.17164 


.98516 


7 


54 


.10279 


99470 


.12014 


99276 


•13744 


99051 


•i547i 


98796 


.17193 


.98511 


6 


55 


. 10308 


99467 


.12043 


99272 


•13773 


99047 


.15500 


98791 


.17222 


.98506 


5 


56 


•10337 


99464 


.12071 


99269 


.13802 


99043 


.15529 


98787 


.17250 


.98501 


4 


57 


. 10366 


9946i 


.12100 


99265 


.13831 


99039 


•15557 


98782 


.17279 


.98496 


3 


58 


• 10395 


99458 


.12129 


99262 


.13860 


99035 


.15586 


98778 


.17308 


.98491 


2 


59 


.10424 


99455 


.12158 


99258 


.13889 


9903 1 


.15615 


98773 


•17336 


.98486 


1 


60 


• 10453 


99452 


.12187 


99255 


■I39I7 


99027 


.15643 


98769 


•17365 


.98481 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


1 

t 


84° 


83 


82 c 


> 


8i c 




8< 


>° 



Natural Sines and Cosines. 



25 



r 


IO 





II 





12° 


13 


D 


14° 


t 


Sine C 


osine 


Sine C 


osine 


Sine 


Cosine 


Sine C 


osine 


Sine C 


osine 


o 


•17365 


98481 


.19081 


98163 


.20791 


.97815 


•22495 


97437 


.24192 


97030 


60 


I 


•17393 


98476 


. 19109 


98157 


.20820 


.97809 


.22523 


97430 


,24220 


97023 


59 


2 


.17422 


98471 


.19138 


98152 


.20848 


.97803 


.22552 


97424 


.24249 


97015 


58 


3 


•17451 


98466 


.19167 


98146 


.20877 


•97797 


.22580 


97417 


.24277 


97008 


57 


4 


•17479 


98461 


•19195 


98140 


.20905 


.97791 


.22608 


97411 


•24305 


97001 


56 


5 


.17508 


98455 


.19224 


98135 


.20933 


•97784 


.22637 


97404 


•24333 


96994 


55 


6 


•17537 


98450 


.19252 


98129 


.20962 


.97778 


.22665 


97398 


.24362 


96987 


54 


7 


•17565 


98445 


.19281 


98124 


. 20990 


.97772 


.22693 


97391 


.24390 


96980 


53 


8 


•17594 


98440 


.19309 


98118 


.21019 


.97766. 


.22722 


97384 


.24418 


96973 


52 


9 


.17623 


98435 


.19338 


98112 


.21047 


.97760 


.22750 


97378 


.24446 


96966 


5i 


IO 


.17651 


98430 


.19366 


98107 


.21076 


•97754 


.22778 


97371 


.24474 


96959 


50 


ii 


.17680 


98425 


•19395 


98101 


.21104 


.97748 


.22807 


97365 


•24503 


96952 


49 


12 


.17708 


98420 


.19423 


98096 


.21132 


.97742 


•22835 


97358 


•24531 


96945 


48 


*3 


•17737 


98414 


.19452 


98090 


.21161 


•97735 


.22863 


97351 


•24559 


96937 


47 


14 


.17766 


98409 


.19481 


98084 


.21189 


.97729 


.22892 


97345 


.24587 


96930 


46 


15 


•17794 


98404 


.19509 


98079 


.21218 


•97723 


.22920 


97338 


.24615 


96923 


45 


16 


.17823 


98399 


.19538 


98073 


.21246 


•97717 


.22948 


97331 


.24644 


96916 


44 


17 


.17852 


98394 


.19566 


98067 


.21275 


.97711 


.22977 


97325 


.24672 


96909 


43 


18 


.17880 


98389 


.19595 


98061 


.21303 


•97705 


•23005 


973i8 


.24700 


96902 


42 


*9 


.17909 


98383 


.19623 


98056 


.21331 


.97698 


.23033 


973 1 1 


.24728 


96894 


41 


20 


•17937 


98378 


.19652 


98050 


.21360 


.97692 


.23062 


97304 


.24756 


96887 


40 


21 


.17966 


98373 


.19680 


98044 


.21388 


.97686 


. 23090 


97298 


.24784 


96880 


39 


22 


•17995 


98368 


• 19709 


98039 


.21417 


.97680 


.23118 


97291 


.24813 


96873 


38 


2 3 


.18023 


98362 


•19737 


98033 


•21445 


•97673 


.23146 


97284 


.24841 


96866 


37 


24 


.18052 


98357 


.19V 66 


98027 


•21474 


.97667 


•23175 


97278 


.24869 


96858 


36 


25 


.18081 


98352 


.19794 


98021 


.21502 


.97661 


.23203 


97271 


.24897 


96851 


35 


26 


.18109 


98347 


.19823 


98016 


.21530 


•97655 


.23231 


97264 


.24925 


96844 


34 


27 


.18138 


98341 


.19851 


98010 


.21559 


.97648 


.23260 


97257 


.24954 


96837 


33 


28 


.18166 


98336 


.19880 


98004 


.21587 


.97642 


.23288 


97251 


.24982 


96829 


32 


29 


.18195 


98331 


. 19908 


97998 


.21616 


.97636 


.23316 


97244 


.25010 


96822 


3i 


30 


.18224 


98325 


.19937 


97992 


.21644 


.97630 


.23345 


97237 


.25038 


96815 


30 


3 1 


.18252 


98320 


.19965 


97987 


.21672 


•97623 


.23373 


97230 


.25066 


96807 


29 


32 


.18281 


98315 


.19994 


97981 


.21701 


•97617 


.23401 


97223 


.25094 


96800 


28 


33 


• 18309 


98310 


.20022 


97975 


.21729 


.97611 


.23429 


97217 


.25122 


96793 


27 


34 


.18338 


98304 


.20051 


97969 


.21758 


.97604 


•23458 


97210 


•25151 


96786 


26 


35 


.18367 


98299 


.20079 


97963 


.21786 


•97598 


.23486 


97203 


•25179 


96778 


25 


36 


.18395 


98294 


.20108 


97958 


.21814 


•97592 


•23514 


97196 


.25207 


96771 


24 


37 


.18424 


98288 


.20136 


97952 


.21843 


•97585 


•23542 


97189 


•25235 


96764 


23 


38 


.18452 


98283 


.20165 


97946 


.21871 


•97579 


•23571 


97182 


.25263 


96756 


22 


39 


.18481 


98277 


.20193 


97940 


.21899 


•97573 


•23599 


97176 


.25291 


96749 


21 


40 


.18509 


98272 


.20222 


97934 


.21928 


.97566 


.23627 


97169 


.25320 


96742 


20 


4i 


.18538 


98267 


.20250 


97928 


.21956 


.97560 


.23656 


97162 


.25348 


96734 


iQ 


42 


.18567 


98261 


.20279 


97922 


.21985 


•97553 


.23684 


97155 


.25376 


96727 


18 


43 


.18595 


98256 


.20307 


979i6 


.22013 


•97547 


.23712 


97148 


.25404 


96719 


17 


44 


.18624 


98250 


.20336 


97910 


.22041 


•97541 


.23740 


97141 


•25432 


96712 


16 


45 


.18652 


98245 


.20364 


97905 


.22070 


•97534 


.23769 


97134 


.25460 


96705 


15 


46 


.18681 


98240 


.20393 


97899 


.22098 


.97528 


•23797 


97127 


.25488 


96697 


14 


47 


.18710 


98234 


.20421 


97893 


.22126 


•97521 


.23825 


97120 


.25516 


96690 


13 


48 


.18738 


98229 


.20450 


97887 


.22155 


•97515 


•23853 


97"3 


•25545 


96682 


12 


49 


.18767 


98223 


.20478 


97881 


.22183 


•975o8 


.23882 


97106 


•25573 


96675 


11 


5o 


•18795 


98218 


.20507 


97875 


.22212 


•97502 


.23910 


97100 


.25601 


96667 


10 


5i 


.18824 


98212 


•20535 


97869 


.22240 


.97496 


.23938 


97093 


.25629 


96660 


9 


52 


.18852 


98207 


.20563 


97863 


.22268 


.97489 


.23966 


97086 


•25657 


96653 


8 


53 


.18881 


98201 


.20592 


97857 


.22297 


•97483 


•23995 


97079 


.25685 


96645 


7 


54 


.18910 


98196 


.20620 


97851 


.22325 


.97476 


.24023 


97072 


•25713 


96638 


6 


55 


.18938 


98190 


. 20649 


97845 


•22353 


•9747o 


.24051 


97065 


•25741 


96630 


5 


56 


.18967 


98185 


.20677 


97839 


.22382 


•97463 


.24079 


97058 


.25769 


96623 


4 


57 


• 18995 


98179 


. 20706 


97833 


.22410 


•97457 


.24108 


97051 


•2579S 


96615 


3 


58 


19024 


98174 


•20734 


97827 


.22438 


•9745° 


•24136 


97044 


.25826 


96608 


2 


59 


.19052 


98168 


•20763 


97821 


.22467 


.97444 


.24164 


97037 


•25854 


96600 


1 


60 


. 19081 


98163 


.20791 


97815 


.22495 


•97437 


.24192 


97030 


.25882 


96593 







Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


79' 


1 


78 


i 


7 


7° 


76 


3 


75' 


) 



26 



Natural Sines and Cosines. 



— 
1 


I 


5° 


1 6° 


I 


7° 


18° 


19° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


.25882 


•96593 


•27564 


.96126 


.29237 


.9563c 


. 30902 


•95*06 


•32557 


.94552 


60 


I 


.25910 


.96585 


.27592 


.96118 


.29265 


.95622 


.30929 


.95097 


•32584 


•94542 


59 


3 


•25938 


.96578 


.27620 


.96110 


.29293 


•95613 


•30957 


.95088 


.32612 


■94533 


58 


3 


.25966 


.96570 


.27648 


.96102 


.29321 


•95605 


•30985 


•95079 


.32639 


•94523 


57 


4 


•25994 


.96562 


.27676 


.96094 


.29348 


•95596 


.31012 


.95070 


.32667 


•945*4 


56 


I 


.26022 


•96555 


.27704 


.96086 


•29376 


•95588 


.31040 


.95061 


.32694 


•94504 


55 


. 26050 


•96547 


.27731 


.96078 


.29404 


•95579 


.31068 


•95052 


.32722 


•94495 


54 


7 


.26079 


.96540 


•27759 


. 96070 


.29432 


•95571 


•3 io 95 


•95043 


•32749 


•94485 


53 


8 


.26107 


•96532 


.27787 


.96062 


.29460 


•95562 


.3**23 


•95033 


•32777 


•94476 


53 


9 


.26135 


.96524 


.27815 


.96054 


•29487 


•95554 


•3"5i 


•95024 


.32804 


.94466 


5* 


IO 


.26163 


•96517 


•27843 


.96046 


•29515 


•95545 


.31178 


•950*5 


.32832 


•94457 


50 


ii 


.26191 


.96509 


.27871 


•96037 


.29543 


.95536 


.31206 


.95006 


•32859 


.94447 


49 


12 


.26219 


.96502 


.27899 


.96029 


•29571 


•95528 


•3 I2 33 


•94997 


.32887 


•94438 


48 


13 


.26247 


.96494 


.27927 


.96021 


.29599 


•95519 


.31261 


.94988 


•32914 


.94428 


47 


14 


.26275 


.96486 


•27955 


•96013 


.29626 


•955" 


.31289 


•94979 


.32942 


.944*8 


46 


15 


.26303 


.96479 


.27983 


.96005 


.29654 


•95502 


.313*6 


•94970 


.32969 


.94409 


45 


16 


•26331 


.96471 


.28011 


•95997 


.29682 


•95493 


•31344 


.94961 


•32997 


•94399 


44 


17 


•26359 


.96463 


.28039 


.95989 


.29710 


•95485 


•31372 


•94952 


.33024 


.94390 


43 


18 


•26387 


.96456 


.28067 


•9598i 


•29737 


•95476 


•31399 


•94943 


•33051 


.94380 


42 


i9 


•26415 


.96448 


.28095 


•95972 


•29765 


•95467 


•3 X 427 


•94933 


•33079 


•94370 


4i 


20 


•26443 


.96440 


.28123 


•95964 


•29793 


•95459 


.31454 


•94924 


.33106 


.94361 


40 


21 


.26471 


•96433 


.28150 


•95956 


.29821 


.95450 


.31482 


•949*5 


•33*34 


•9435* 


39 


22 


.26500 


.96425 


.28178 


.95948 


.29849 


•95441 


•3*5*o 


.94906 


.33161 


•94342 


38 


23 


.26528 


.96417 


.28206 


.95940 


.29876 


•95433 


•31537 


.94897 


•33*89 


•94332 


37 


24 


.26556 


.96410 


.28234 


•95931 


.29904 


.95424 


•3*565 


.94888 


.33216 


.94322 


36 


25 


• 26584 


. 96402 


.28262 


•95923 


.29932 


•95415 


•3*593 


.94878 


•33244 


•943*3 


35 


26 


.26612 


.96304 


.28290 


•95915 


.29960 


•95407 


.31620 


.94869 


•33271 


•94303 


34 


27 


. 26640 


.96386 


.28318 


•959°7 


.29987 


•95398 


.31648 


.94860 


.33298 


•94293 


33 


28 


.26668 


•96379 


.28346 


.95898 


.30015 


•95389 


•3*675 


.94851 


.33326 


.94284 


32 


29 


.26696 


•96371 


.28374 


•95890 


•30043 


•9538o 


•3*703 


.94842 


•33353 


.94274 


3* 


30 


.26724 


•96363 


.28402 


.95882 


.30071 


•95372 


•3*730 


•94832 


.3338i 


.94264 


30 


31 


.26752 


•96355 


.28429 


•95874 


. 30098 


•95363 


.31758 


•94823 


•334o8 


•94254 


29 


32 


.26780 


•96347 


•28457 


.95865 


.30126 


•95354 


.3*786 


.94814 


•33436 


•94245 


28 


33 


.26808 


.96340 


.28485 


.95857 


•30154 


•95345 


•3*8*3 


•94805 


•33463 


■94235 


37 


34 


.26836 


•96332 


.28513 


.95849 


.30182 


•95337 


•3*841 


•94795 


■3349° 


•94225 


26 


35 


.26864 


.96324 


.28541 


.95841 


. 30209 


•95328 


.31868 


.94786 


•335*3 


•94215 


25 


36 


.26892 


.96316 


.28569 


.95832 


•30237 


•95319 


.31896 


•94777 


•33545 


.94206 


24 


37 


.26920 


.96308 


•28597 


.95824 


.30265 


•953io 


•3*923 


.94768 


•33573 


.94196 


23 


38 


.26948 


.96301 


.28625 


.95816 


.30292 


•953oi 


•3*95* 


•94758 


.33600 


.94186 


22 


39 


.26976 


•96293 


.28652 


.95807 


.30320 


•95293 


•3*979 


.94749 


•33627 


•94*76 


21 


40 


.27004 


.96285 


.28680 


•95799 


•30348 


.95284 


.32006 


•9474° 


•33655 


•94*67 


20 


4i 


.27032 


.96277 


.28708 


•95791 


.30376 


•95275 


•32034 


•9473° 


.33682 


•94*57 


*9 


42 


.27060 


.96269 


•28736 


•95782 


.30403 


.95266 


.32061 


.94721 


•337*0 


•94*47 


18 


43 


.27088 


.96261 


.28764 


•95774 


•30431 


•95257 


.32089 


•947*2 


•33737 


•94*37 


*7 


44 


.27116 


•96253 


.28792 


•95766 


• 30459 


.95248 


.32116 


.94702 


•33764 


•94*27 


16 


45 


.27144 


.96246 


.28820 


•95757 


. 30486 


.95240 


.32144 


•94693 


•33792 


.94118 


15 


46 


.27172 


.96238 


.28847 


•95749 


• 305*4 


.95231 


.32171 


.94684 


•338i9 


.94108 


14 


47 


.27200 


.96230 


.28875 


•95740 


• 30542 


.95222 


.32109 


.94674 


.33846 


.94098 


*3 


48 


.27228 


.96222 


.28903 


.95732 


•30570 


.95213 


.32227 


•94665 


•33874 


.94088 


12 


49 


.27256 


.96214 


.2893! 


•95724 


•30597 


.95204 


•32254 


.94656 


.33901 


.94078 


11 


5° 


.27284 


.96206 


.28959 


•95715 


.30625 


•95195 


.32282 


.94646 


•33929 


.94068 


10 


5i 


.27312 


.96198 


.28987 


•95707 


•30653 


.95186 


•32309 


•94637 


•33956 


.94058 


9 


52 


.27340 


.96190 


.29015 


.95698 


. 30680 


•95177 


•32337 


.94627 


•33983 


•94049 


8 


53 


•27368 


.96182 


.29042 


.95690 


.30708 


.95168 


•32364 


.94618 


.34011 


•94039 


7 


54 


.27396 


.96174 


.29070 


.95681 


•30736 


•95159 


.32392 


.94609 


.34038 


.94029 


6 


55 


.27424 


.96166 


.29098 


•95673 


•30763 


•95150 


•324*9 


•94599 


•34065 


.94019 


5 


56 


•27452 


.96158 


.29126 


.95664 


•30791 


.95142 


•32447 


•94590 


•34093 


. 94009 


4 


57 


.27480 


.96150 


.29154 


.95656 


•30819 


•95133 


•32474 


.94580 


.34*20 


•93999 


3 


58 


.27508 


.96142 


.29182 


•95647 


.30846 


•95* 2 4 


.32502 


•9457* 


•34*47 


.93989 


2 


59 


•27536 


.96134 


.29209 


•95639 


•30874 


•95**5 


•32529 


.94561 


•34*75 


•93979 


1 


60 


•27564 


.96126 


.29237 


•95630 


. 30002 


.95106 


•32557 


•94552 


.34202 


•93969 





1 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


I 


7 


4° ! 


73 





7 


2° 


71 





7c 






Natural Sines and Cosines. 



27 



f 


20° 


21° 


22° 


23 


3 


24° 


t 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


osine 


Sine ( 


Cosine 


o 


.34202 


.93969 


.35837 


•93358 


•3746r 


.92718 


•39073 


92050 


.40674 


•91355 


60 


I 


.34229 


•93959 


.35864 


•93348 


.37488 


.92707 


.39100 


92039 


.40700 


•91343 


59 


2 


•34257 


•93949 


.35891 


•93337 


•37515 


.92697 


.39127 


92028 


.40727 


•9133 1 


58 


3 


.34284 


•93939 


.35918 


•93327 


•37542 


.92686 


•39153 


92016 


•40753 


•91319 


57 


4 


•343 11 


.93929 


•35945 


.933i6 


•37569 


.92675 


.39180 


92005 


.40780 


•91307 


56 


5 


•34339 


.93919 


•35973 


.93306 


•37595 


.92664 


•39207 


91994 


. 40806 


.91295 


55 


6 


.34366 


•93909 


.36000 


•93295 


.37622 


•92653 


•39234 


91982 


•40833 


.91283 


54 


7 


•34393 


.93899 


.36027 


.93285 


•37649 


.92642 


.39260 


91971 


. 40860 


.91272 


53 


8 


•34421 


.93889 


•3 6 °54 


•93274 


.37676 


.92631 


.39287 


91959 


.40886 


.91260 


52 


9 


•34448 


•93879 


.36081 


.93264 


•37703 


.92620 


•39314 


91948 


.40913 


.91248 


5i 


IO 


•34475 


.93869 


.36108 


•93253 


•37730 


.92609 


•3934i 


91936 


•40939 


.91236 


50 


ii 


•34503 


•93859 


.36135 


•93243 


•37757 


.92598 


•39367 


9*925 


.40966 


.91224 


4Q 


12 


•34530 


.93849 


.36162 


.93232 


•37784 


.92587 


•39394 


91914 


.40992 


.91212 


48 


13 


•34557 


•93839 


.36190 


.93222 


.37811 


.92576 


.39421 


91902 


.41019 


.91200 


47 


14 


.34584 


.93829 


.36217 


.93211 


.37838 


•92565 


•39448 


91891 


.41045 


.91188 


46 


15 


.34612 


.93819 


.36244 


.93201 


.37865 


•92554 


•39474 


91879 


.41072 


.91176 


45 


16 


• 34639 


.93809 


.36271 


.93190 


•37892 


•92543 


•395 QI 


91868 


.41098 


.91164 


44 


17 


. 34666 


•93799 


.36298 


.93180 


•379 I 9 


•92532 


•39528 


91856 


.41125 


.91152 


43 


18 


.34694 


•93789 


.36325 


.93169 


•37946 


.92521 


•39555 


9^45 


.41151 


.91140 


42 


19 


•34721 


•93779 


•36352 


•93159 


•37973 


.92510 


.3958i 


91833 


.41178 


.91128 


41 


20 


•34748 


•93769 


•36379 


.93148 


•37999 


.92499 


.39608 


91822 


.41204 


.91116 


40 


21 


•34775 


•93759 


.36406 


•93137 


.38026 


.92488 


•39635 


91810 


.41231 


.91104 


39 


22 


•34803 


•93748 


•36434 


•93 I2 7 


.38053 


.92477 


.39661 


91799 


•41257 


.91092 


38 


23 


• 34830 


•93738 


.36461 


.93116 


.38080 


.92466 


.39688 


91787 


.41284 


.91080 


37 


24 


•34857 


.93728 


.36488 


.93106 


.38107 


.92455 


.39715 


9*775 


.41310 


.91068 


36 


25 


.34884 


.937i8 


.36515 


•93095 


.38i34 


.92444 


•39741 


91764 


•41337 


.91056 


35 


26 


.34912 


.93708 


.36542 


.93084 


.38161 


.92432 


.39768 


91752 


•41363 


.91044 


34 


27 


•34939 


.93698 


.36569 


•93074 


.38188 


.92421 


•39795 


91741 


.41390 


.91032 


33 


28 


.34966 


.93688 


.36596 


•93063 


.38215 


.92410 


.39822 


91729 


.41416 


.91020 


32 


29 


•34993 


•93677 


.36623 


•93052 


.38241 


.92399 


•39848 


91718 


•41443 


.91008 


31 


3° 


.35021 


.93667 


.36650 


.93042 


.38268 


.92388 


•39875 


91706 


.41469 


. 90996 


30 


31 


.35048 


•93657 


.36677 


.93031 


.38295 


•92377 


.39902 


91694 


.4149 6 


.90984 


29 


32 


•35075 


•93647 


.36704 


.93020 


.38322 


.92366 


.39928 


91683 


.41522 


.90972 


28 


33 


•35 io 2 


•93637 


.36731 


.93010 


.38349 


•92355 


•39955 


91671 


•41549 


. 90960 


27 


34 


•3513° 


.93626 


.36758 


.92999 


.38376 


•92343 


•39982 


91660 


•41575 


.90948 


26 


35 


•35157 


.93616 


.36785 


.92988 


.38403 


.92332 


. 40008 


91648 


.41602 


.90936 


25 


36 


.35184 


.93606 


.36812 


.92978 


.38430 


.92321 


•40035 


91636 


.41628 


.90924 


24 


37 


•352II 


•93596 


.36839 


.92967 


.38456 


.92310 


.40062 


91625 


•41655 


.90911 


23 


38 


•35239 


•93585 


.36867 


.92956 


.38483 


.92299 


.40088 


91613 


.41681 


.90899 


22 


39 


.35266 


•93575 


.36894 


•92945 


.38510 


.92287 


.40115 


91601 


•41707 


.90887 


21 


40 


•35293 


•93565 


.36921 


•92935 


.38537 


.92276 


.40141 


9 I 59° 


•41734 


.90875 


20 


4i 


•35320 


•93555 


.36948 


.92924 


.38564 


.92265 


.40168 


91578 


.41760 


.90863 


19 


42 


•35347 


•93544 


.36975 


.92913 


.38591 


.92254 


.40195 


91566 


.41787 


.90851 


18 


43 


•35375 


•93534 


.37002 


.92902 


.38617 


.92-43 


.40221 


91555 


.41813 


.90839 


17 


44 


•35402 


•93524 


.37029 


.92892 


.38644 


.92231 


.40248 


91543 


.41840 


.90826 


16 


45 


•35429 


•935H 


.37056 


.92881 


.38671 


.92220 


•40275 


9i53i 


.41866 


.90814 


i5 


46 


•35456 


•93503 


.37083 


.92870 


.38698 


.92209 


.40301 


91519 


.41892 


.90802 


14 


47 


•35484 


•93493 


.37110 


.92859 


.38725 


.92198 


.40328 


91508 


.41919 


.90790 


13 


48 


•355" 


•93483 


•37137 


.92849 


•38752 


.92186 


•40355 


91496 


•41945 


.90778 


12 


49 


•35538 


•93472 


.37164 


.92838 


.38778 


•92175 


.40381 


91484 


.41972 


.90766 


11 


5o 


•35565 


.93462 


.37191 


.92827 


.38805 


.92164 


.40408 


91472 


.41998 


•90753 


10 


51 


•35592 


•93452 


.37218 


.92816 


.38832 


.92152 


•40434 


91461 


.42024 


.90741 


9 


52 


•356i9 


•93441 


•37245 


.92805 


.38859 


.92141 


.40461 


91449 


.42051 


.90729 


8 


53 


•35647 


•93431 


.37272 


.92794 


.38886 


.92130 


.40488 


91437 


.42077 


.90717 


7 


54 


•35674 


.93420 


•37299 


.927C4 


.38912 


.92119 


.40514 


91425 


.42104 


.90704 


6 


55 


•35701 


.93410 


•37326 


•92773 


•38939 


.92107 


.40541 


91414 


.42130 


. 90692 


5 


56 


.35728 


.93400 


•37353 


.92762 


.38966 


.92096 


.40567 


91402 


.42156 


. 000S0 


4 


57 


•35755 


•93389 


•3738o 


.92751 


•38993 


.92085 


.40594 


91390 


.42183 


.90668 


3 


58 


•35782 


•93379 


• 37407 


.92740 


.39020 


.92073 


.40621 


9137S 


.42209 


•90655 


2 


§ 9 


.35810 


.93368 


•37434 


.92729 


.39046 


.92062 


.40647 


91366 


.42235 


.90643 


X 


60 


•35837 


•93358 


•3746i 


.92718 


• 39073 


.92050 


.40674 


91355 


.42262 


.00631 





f 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


6 


9° 


6 


5° 


6 


7° 1 


66 


3 


65 






28 



Natural Sines and Cosines. 



- 


*S° 


26 


O 


27 


O 


28 





29 





.1 


Sine 


Cosine 


Sine ( 


Cosine 


Sine ( 


Cosine 


Sine ( 


Cosine 


Sine Cosine 


o 


.42262 


.90631 


•43837 


.89879 


•45399 


.89101 


•46947 


.88295 


.484S1 


.87462 


60 


I 


.42288 


.90618 


.43863 


.89867 


•45425 


.89087 


•46973 


.88281 


.48506 


.87448 


59 


2 


•42315 


.90606 


.43889 


.89854 


.45451 


.80074 


.46999 


.88267 


.48532 


•87434 


58 


3 


.42341 


.90594 


.43916 


.89841 


•45477 


.89061 


•47024 


.88254 


.48557 


.87420 


57 


4 


.42367 


.90582 


•43942 


.89828 


•45503 


.89048 


.47050 


.88240 


.48583 


.87406 


56 


5 


•42394 


.90569 


.43968 


.89816 


•45529 


.89035 


.47076 


.88226 


.48608 


•87391 


55 


6 


.42420 


•90557 


•43994 


.89803 


•45554 


.89021 


.47101 


.88213 


.48634 


.87377 


54 


7 


.42446 


•90545 


.44020 


.89790 


•4558o 


.89008 


.47127 


.88199 


.48659 


.87363 


53 


8 


•42473 


.90532 


. 44046 


.89777 


.45606 


.88995 


•47153 


.88185 


.48684 


•87349 


5* 


9 


•42499 


.90520 


.44072 


.89764 


•45632 


.88981 


.47178 


.88172 


.48710 


•87335 


5i 


IO 


•42525 


.90507 


.44098 


.89752 


•45658 


.88968 


.47204 


.88158 


.48735 


•87321 


50 


ii 


•42552 


.90495 


.44124 


•89739 


.45684 


.88955 


.47229 


.88144 


.48761 


.87306 


49 


12 


.42578 


.90483 


•44 I 5i 


.89726 


•457io 


.88942 


•47255 


.88130 


.48786 


.87292 


48 


13 


.42604 


.90470 


•44*77 


•89713 


•45736 


.88928 


.47281 


.88117 


.48811 


.87278 


47 


14 


.42631 


.90458 


.44203 


.89700 


•45762 


.88915 


•47306 


.88103 


.48837 


.87264 


46 


15 


•42657 


.90446 


.44229 


.89687 


.45787 


.88902 


• 4733-2 


.88089 


.48862 


.87250 


45 


l6 


.42683 


•90433 


•44255 


.89674 


.45813 


.88888 


•47358 


•88075 


.48888 


•87235 


44 


17 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


•47383 


.88062 


.48913 


.87221 


43 


18 


•42736 


. 90408 


•44307 


.89649 


.45865 


.88862 


.47409 


.88048 


.48938 


.87207 


42 


19 


.42762 


.90396 


•44333 


.89636 


.45891 


.88848 


•47434 


.88034 


.48964 


•87193 


4i 


20 


.42788 


.90383 


•44359 


.89623 


•45917 


.88835 


.47460 


.88020 


.48989 


.87178 


40 


21 


.42815 


•90371 


•44385 


.89610 


.45942 


.88822 


.47486 


.88006 


.49014 


.87164 


39 


22 


.42841 


.90358 


.44411 


•89597 


.45968 


.88808 


•475II 


•87993 


.49040 


.87150 


38 


23 


.42867 


.90346 


•44437 


.89584 


•45994 


.88795 


•47537 


•87979 


.49065 


.87136 


37 


24 


.42804 


•90334 


.44464 


•89571 


.46020 


.88782 


•47562 


•87965 


.49090 


.87121 


36 


25 


.42920 


.90321 


.44490 


.89558 


. 46046 


.88768 


.47588 


•87951 


•49 116 


.87107 


35 


26 


.42946 


•90309 


.44516 


•89545 


.46072 


•88755 


.47614 


•87937 


.49141 


.87093 


34 


27 


.42972 


.90296 


•44542 


.89532 


.46097 


.88741 


•47639 


•87923 


.49166 


.87079 


33 


28 


•42999 


.90284 


.44568 


.89519 


.46123 


.88728 


.47665 


.87909 


.49192 


.87064 


32 


29 


.43025 


.90271 


•44594 


.89506 


.46149 


.88715 


.47690 


.87896 


.49217 


.87050 


3i 


30 


•43°5i 


.90259 


.44620 


.89493 


•46175 


.88701 


.47716 


.87882 


.49242 


.87036 


30 


31 


•43°77 


.90246 


.44646 


.89480 


.46201 


.88688 


•47741 


.87868 


.49268 


.87021 


29 


32 


.43104 


.90233 


.44672 


.89467 


.46226 


.88674 


.47767 


.87854 


.49293 


.87007 


28 


33 


•43130 


.90221 


.44698 


.89454 


.46252 


.88661 


•47793 


.87840 


.49318 


.86993 


27 


34 


• 43*56 


. 90208 


.44724 


.89441 


.46278 


.88647 


.47818 


.87826 


•49344 


.86978 


26 


35 


.43182 


.90196 


•4475o 


.89428 


.46304 


.88634 


.47844 


.87812 


.49369 


.86964 


25 


36 


.43209 


.90183 


.44776 


.89415 


•46330 


.88620 


.47869 


.87798 


•49394 


.86949 


24 


37 


•43235 


.90171 


. 44802 


.89402 


•46355 


.88607 


•47895 


.87784 


.49419 


.86935 


23 


38 


.43261 


.90158 


.44828 


.89389 


.46381 


•88593 


.47920 


.87770 


•49445 


.86921 


22 


39 


.43287 


.90146 


.44854 


.89376 


.46407 


.88580 


•47946 


•87756 


.49470 


.86906 


21 


40 


•43313 


•90133 


.44880 


.89363 


•46433 


.88566 


•4797 1 


•87743 


•49495 


.86892 


20 


4i 


•43340 


.90120 


. 44906 


.89350 


.46458 


.88553 


•47997 


.87729 


.49521 


.86878 


19 


42 


.43366 


.90108 


•44932 


•89337 


.46484 


.88539 


.48022 


.87715 


•49546 


.86863 


18 


43 


=43392 


.90095 


.44958 


•89324 


.46510 


.88526 


.48048 


.87701 


•4957* 


.86849 


17 


44 


.43418 


.90082 


•44984 


.89311 


•46536 


.88512 


•48073 


.87687 


•49596 


.86834 


16 


45 


•43445 


. 90070 


.45010 


.89298 


•46561 


.88499 


.48099 


.87673 


.49622 


.86820 


15 


46 


43471 


.90057 


•45036 


.89285 


.46587 


.88485 


.48124 


.87659 


•49647 


.86805 


14 


47 


43497 


•90045 


.45062 


.89272 


•46613 


.88472 


.48150 


.87645 


.49672 


.86791 


13 


48 


•43523 


.90032 


.45088 


.89259 


.46639 


.88458 


.48175 


.87631 


•49697 


.86777 


12 


49 


•43549 


.90019 


•45H4 


•89245 


.46664 


.88445 


.48201 


.87617 


•49723 


.86762 


11 


5° 


•43575 


.90007 


.45140 


.89232 


. 46690 


.88431 


.48226 


.87603 


.49748 


.86748 


10 


5i 


.43602 


•89994 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


•49773 


.86733 


9 


52 


.43628 


.89981 


•45192 


.89206 


.46742 


.88404 


.48277 


•87575 


.49798 


.86719 


8 


53 


•43654 


.89968 


.45218 


.89193 


.46767 


.88390 


•48303 


.87561 


.49824 


.86704 


7 


54 


.43680 


.89956 


•45243 


.89180 


•46793 


•88377 


.48328 


.87546 


•49849 


.86690 


6 


55 


.43706 


.89943 


•45269 


.89167 


.46819 


•88363 


•48354 


•87532 


•49874 


.86675 


5 


56 


•43733 


.89930 


•45295 


.89153 


.46844 


•88349 


•48379 


.87518 


•49899 


.86661 


4 


57 


•43759 


.89918 


•45321 


.89140 


.46870 


•88336 


•48405 


•87504 


•49924 


.86646 


3 


58 


•43785 


.89905 


•45347 


.89127 


.46896 


.88322 


.48430 


.87490 


•4995° 


.86632 


2 


59 


.43811 


.89892 


•45373 


.89114 


.46921 


.88308 


•48456 


.87476 


•49975 


.86617 


1 


60 


•43837 


.89879 


•45399 


.89101 


•46947 


.88295 


.48481 


.87462 


.50000 


.86603 





t 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


1 « 


4° 


U 6, 


3° 


6: 


>° 


6] 


<° 


6c 


)° 



Natural Sines and Cosines. 



29 



1- 


30 





31 





32 





33 







34 


f 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine C 


Cosine 


o 


.50000 


.86603 


.51504 


.85717 


.52992 


.84805 


•54464 


.83867 


•55919 


82904 


60 


J 


.50025 


.86588 


•5i5 2 9 


.85702 


.53017 


.84789 


.54488 


.83851 


•55943 


82887 


59 


2 


■ 5°°5° 


.86573 


•51554 


.85687 


.53041 


.84774 


•545*3 


•83835 


.55968 


82871 


58 


3 


,50076 


.86559 


•51579 


.85672 


.53066 


•84759 


•54537 


.83819 


•55992 


82855 


57 


4 


.50101 


.86544 


.51604 


.85657 


•53091 


•84743 


•5456i 


.83804 


.56016 


82839 


56 


5 


.50126 


.86530 


.51628 


.85642 


•53"5 


.84728 


•54586 


.83788 


. 56040 


82822 


55 


6 


•50151 


.86515 


•51653 


.85627 


•53 J 4o 


.84712 


.54610 


.83772 


. 56064 


82806 


54 


7 


.50176 


.86501 


.51678 


.85612 


•53164 


.84697 


•54635 


.83756 


. 56088 


82790 


53 


8 


.50201 


.86486 


•51703 


•85597 


•53189 


.84681 


•54659 


•83740 


.56112 


82773 


52 


9 


.50227 


.86471 


.51728 


.85582 


•53214 


.84666 


•54683 


.83724 


.56136 


82757 


5i 


IO 


.50252 


.86457 


•51753 


.85567 


.53238 


.84650 


.547o8 


.83708 


.56160 


82741 


50 


ii 


.50277 


.86442 


.51778 


.85551 


.53263 


.84635 


•54732 


.83692 


.56184 


82724 


49 


12 


. 50302 


.86427 


.51803 


•85536 


.53288 


.84619 


•54756 


•83676 


.56208 


82708 


48 


!3 


.50327 


.86413 


.51828 


.85521 


.53312 


.84604 


.5478i 


.83660 


.56232 


82692 


47 


14 


•5°3S2 


.86398 


.51852 


.85506 


•53337 


.84588 


•54805 


.83645 


.56256 


82675 


46 


15 


•50377 


.86384 


.51877 


.85491 


•5336i 


.84573 


.54829 


.83629 


.56280 


82659 


45 


l6 


• 50403 


.86369 


.51902 


.85476 


.53386 


.84557 


•54854 


.83613 


•56305 


82643 


44 


*7 


.50428 


.86354 


.51927 


,85461 


•534H 


.84542 


.54878 


•83597 


•56329 


82626 


43 


18 


•50453 


.86340 


•5*952 


.85446 


•53435 


.84526 


• 54902 


.83581 


•56353 


82610 


42 


19 


.50478 


.86325 


.51977 


.85431 


.5346o 


.84511 


•54927 


.83565 


•56377 


82593 


4i 


20 


•50503 


.86310 


.52002 


.85416 


•53484 


.84495 


•54951 


.83549 


. 56401 


82577 


40 


21 


.50528 


.86295 


.52026 


.85401 


•53509 


.84480 


•54975 


.83533 


•56425 


82561 


39 


22 


•50553 


.86281 


.52051 


.85385 


•53534 


.84464 


•54999 


•83517 


•56449 


82544 


38 


23 


•50578 


.86266 


.52076 


.85370 


.53558 


.84448 


.55024 


.83501 


.56473 


82528 


37 


24 


.50603 


.86251 


.52101 


•85355 


.53583 


•84433 


•55048 


.83485 


•56497 


82511 


36 


25 


.50628 


.86237 


.52126 


•85340 


•53607 


•84417 


•55072 


.83469 


.56521 


82495 


35 


26 


•50654 


.86222 


.52151 


.85325 


.53632 


. 84402 


.55097 


•83453 


•56545 


82478 


34 


27 


. 50679 


.86207 


•52T75 


.85310 


•53656 


.84386 


.55121 


•83437 


.56569 


82462 


33 


28 


.50704 


.86152 


.52200 


.85294 


.53681 


.84370 


•55145 


.83421 


•56593 


82446 


32 


29 


•50729 


.86178 


.52225 


•85279 


.53705 


•84355 


•55169 


.83405 


.56617 


82429 


31 


3° 


•50754 


.86163 


.52250 


.85264 


•53730 


.84339 


•55194 


.83389 


.56641 


82413 


30 


31 


•50779 


.86148 


•52275 


.85249 


•53754 


.84324 


.55218 


•83373 


.56665 


82396 


29 


32 


.50804 


.86133 


• 52299 


•85234 


•53779 


.84308 


•55242 


.83356 


.56689 


82380 


28 


33 


.50829 


.86119 


•52324 


.85218 


.53804 


.84292 


.55266 


.83340 


.56713 


82363 


27 


34 


• 50854 


.86104 


.52349 


.85203 


.53828 


.84277 


.55291 


.83324 


•56736 


82347 


26 


35 


.50879 


.86089 


.52374 


.85188 


.53853 


.84261 


•55315 


.83308 


.56760 


82330 


25 


36 


•50904 


.86074 


•52399 


•85173 


.53877 


.84245 


•55339 


.83292 


.56784 


82314 


24 


37 


•50929 


.86059 


•52423 


.85157 


.53902 


.84230 


•55363 


•83276 


.56808 


82297 


23 


38 


•50954 


. 86045 


•52448 


.85142 


.53926 


.84214 


.55388 


.83260 


.56832 


82281 


22 


39 


•50979 


.86030 


•52473 


.85127 


•53951 


.84198 


•55412 


•83244 


.56856 


82264 


21 


40 


.51004 


.86015 


.52498 


.85112 


•53975 


.84182 


•55436 


.83228 


.56880 


82248 


20 


41 


.51029 


.86000 


.52522 


.85096 


. 54000 


.84167 


.5546o 


.83212 


.56904 


82231 


19 


42 


.51054 


•85985 


•52547 


.85081 


.54024 


.84151 


.55484 


•83195 


.56928 


82214 


18 


43 


•5 T °79 


.85970 


•52572 


.85066 


•54049 


.84135 


.55509 


.83179 


•56952 


82198 


17 


44 


.51104 


.85956 


•52597 


.85051 


• 54073 


.84120 


•55533 


.83163 


• 56976 


82181 


16 


45 


.51129 


.85941 


.52621 


.85035 


•54097 


.84104 


•55557 


.83147 


.57000 


82165 


15 


46 


•5"54 


.85926 


.52646 


.85020 


.54122 


.84088 


.55581 


.83131 


.57024 


82148 


14 


47 


•5"79 


.85011 


.52671 


•85005 


.54146 


.84072 


•55605 


.83115 


•57047 


82132 


13 


48 


051204 


.85896 


.52696 


.84989 


•54i7i 


•84057 


•55630 


.83098 


•57071 


8211.5 


12 


4Q 


.51229 


.85881 


.52720 


.84974 


•54195 


.84041 


•55654 


.83082 


•57095 


82098 


11 


50 


•5 I 254 


.85866 


•52745 


.84959 


.54220 


.84025 


.55678 


.83066 


•57II9 


82082 


10 


5i 


•51279 


.85851 


.52770 


.84943 


• 54244 


. 84009 


• 55702 


.83050 


.57M3 


82065 


9 


52 


.51304 


.85836 


•52794 


.84928 


.54269 


•83994 


.55726 


.83034 


•57167 


82048 


8 


53 


•51329 


.85821 


.52819 


.84913 


•54293 


.83978 


.55750 


.83017 


.S7191 


82032 


7 


54 


•51354 


.85806 


•52844 


.84897 


•54317 


.83962 


•55775 


.83001 


•57215 


820*5 


6 


55 


•51379 


.85792 


.52869 


.84882 


•54342 


.83946 


•55799 


.82985 


•57238 


81999 


5 


56 


.51404 


•85777 


•52893 


.84866 


•54366 


.83930 


•55823 


.82969 


.57262 


81982 


4 


57 


•51429 


.85762 


.52918 


.84851 


•5439 1 


•83915 


'55847 


.82953 


•57286 


81965 


3 


58 


•51454 


•85747 


•52943 


.84836 


•54415 


.83899 


•55871 


.82936 


•573 10 


81949 


2 


59 


•5M79 


•85732 


.52967 


.84820 


•54440 


.83883 


•55895 


.82920 


•57334 


81932 


1 


60 


•51504 


•85717 


.52992 


.84805 


.54464 


.83867 


•55919 


.82004 


•57358 


81913 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


1 


59 





58 





I 57 





56 





55' 


1 



30 



Natural Sines and Cosines. 



/ 


35 





36 


37° 


38° 


39' 


t 


Sine 


Cosine 


Sine C 


osine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


•57358 


.81915 


.58779 


80902 


.60182 


.79864 


.61566 


.78801 


.62932 


.77715 


60 


I 


.57381 


.81899 


.58802 


80885 


.6c 205 


.79846 


.61589 


.78783 


•62955 


.77696 


59 


2 


•57405 


,81882 


.58826 


80867 


.60228 


.79829 


.61612 


.78765 


.62977 


.77678 


58 


3 


•57429 


.81865 


.58849 


80850 


.60251 


.79811 


.61635 


•78747 


.63000 


.77660 


57 


4 


•57453 


.81848 


•58873 


80833 


.60274 


•79793 


.61658 


.78729 


.63022 


.7764* 


56 


5 


•57477 


.81832 


.58896 


80816 


.60298 


.79776 


.61681 


.78711 


.63045 


.77623 


55 


6 


•57501 


.81815 


.58920 


80799 


.60321 


.79758 


.61704 


.78694 


.63068 


.77605 


54 


7 


•57524 


.81798 


•58943 


80782 


.60344 


•79741 


.61726 


.78676 


.63090 


.77586 


53 


8 


•57548 


.81782 


.58967 


80765 


.60367 


•79723 


.61749 


.78658 


.63113 


.77568 


52 


9 


•57572 


.81765 


.58990 


80748 


.60390 


.79706 


.61772 


.78640 


•63135 


•7755o 


5i 


IO 


•57596 


.81748 


•59014 


80730 


.60414 


.79688 


•61795 


.78622 


.63158 


•7753i 


5° 


„ 


.57619 


.81731 


•59037 


80713 


.60437 


.79671 


.61818 


. 78604 


.63180 


•77513 


49 


12 


•57643 


.81714 


.59061 


80696 


.60460 


•79653 


.61841 


.78586 


.63203 


•77494 


48 


13 


.57667 


.81698 


.59084 


80679 


.60483 


•79635 


.61864 


.78568 


.63225 


•77476 


47 


14 


•57691 


.81681 


.59108 


80662 


.60506 


.79618 


.61887 


.78550 


.63248 


•77458 


46 


15 


•577*5 


.81664 


•59I3 1 


80644 


.60529 


.79600 


.61909 


.78532 


.63271 


•77439 


45 


16 


.57738 


.81647 


.59154 


80627 


•60553 


•79583 


.61932 


.78514 


.63293 


•77421 


44 


17 


•57762 


.81631 


.59178 


80610 


.60576 


•79565 


•61955 


.78496 


.63316 


.77402 


43 


18 


.57786 


.81614 


.50201 


80593 


.60599 


•79547 


.61978 


.78478 


•63338 


•77384 


42 


19 


.57810 


.81597 


•59225 


80576 


.60622 


.79530 


.62001 


.78460 


.63361 


.77366 


4i 


20 


.57833 


.81580 


.59248 


80558 


.60645 


•79512 


.62024 


.78442 


.63383 


•77347 


40 


21 


.57857 


.81563 


.59272 


80541 


.60668 


.79494 


. 62046 


.78424 


. 63406 


.77329 


39 


22 


.57881 


.81546 


.59295 


80524 


.60691 


•79477 


. 62069 


.78405 


.63428 


•773 I o 


38 


23 


.57904 


.81530 


.59318 


: V-7 


.60714 


•79459 


. 62092 


.78387 


•63451 


.77292 


3 l 


24 


.57928 


.81513 


•59342 


804S9 


•60738 


.79441 


.62115 


■78369 


.63473 


.77273 


36 


25 


•57952 


.81496 


•59365 


80472 


.60761 


.79424 


.62138 


•78351 


.63496 


•77255 


35 


26 


•57976 


.81479 


.59389 


8o455 


.60784 


.79406 


.62160 


•78333 


.63518 


.77236 


34 


27 


•57999 


.81462 


.59412 


80438 


.60807 


.79388 


.62183 


.78315 


.63540 


.77218 


33 


28 


.58023 


.81445 


•59436 


80420 


.60830 


•79371 


.62206 


.78297 


•63563 


.77109 


32 


29 


.58047 


.81428 


•59459 


80403 


•60853 


•79353 


.62229 


.78279 


•63585 


.77i8i 


3i 


3° 


.58070 


.81412 


.59482 


80386 


.60876 


•79335 


.62251 


.78261 


.63608 


.77162 


30 


3i 


.58094 


.81395 


.595o6 


80368 


.60899 


.793i8 


.62274 


.78243 


.63630 


.77144 


29 


32 


.58118 


.81378 


•59529 


80351 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


28 


33 


.58141 


.81361 


•59552 


80334 


.60945 


.79282 


.62320 


.78206 


.63675 


.77107 


27 


34 


.58165 


.81344 


•59576 


80316 


.60968 


. 79264 


.62342 


.78188 


.63698 


.7708S 


26 


35 


.58189 


.81327 


•59599 


80299 


.60991 


.79247 


.62365 


.78170 


.63720 


.77070 


25 


36 


.58212 


.81310 


.59622 


80282 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


24 


37 


.58236 


.81293 


.59646 


80264 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


23 


38 


.58260 


.81276 


.59669 


80247 


.61061 


•79 T 93 


•62433 


.78116 


.63787 


.77014 


22 


39 


.58283 


.81259 


•59603 


80230 


.61084 


.79176 


.62456 


.78098 


.63810 


.76996 


21 


40 


•58307 


.81242 


•597i6 


80212 


.61107 


•79158 


.62479 


.78079 


.63832 


•76977 


20 


4i 


.58330 


.81225 


•59739 


80195 


.61130 


.79140 


.62502 


.78061 


.63854 


.76959 


19 


42 


.58354 


.81208 


•59763 


80178 


.61153 


.79122 


.62524 


.78043 


.63877 


. 76940 


18 


43 


•58378 


.81191 


.59786 


80160 


.61176 


•79 io 5 


.62547 


.78025 


.63899 


.76921 


17 


44 


.58401 


.81174 


.59809 


80143 


.61199 


.79087 


.62570 


.78007 


.63922 


•76903 


16 


45 


58425 


.81157 


•59832 


80125 


.61222 


. 79069 


.62592 


.77988 


.63944 


.76884 


15 


46 


,58449 


.81140 


.59856 


80108 


.61245 


•79051 


.62615 


•77970 


.63966 


. 76866 


14 


47 


•58472 


.81123 


•59879 


80091 


.61268 


•79033 


.62638 


.77952 


.63980 


.76847 


13 


48 


.58496 


.81106 


.59902 


80073 


.61291 


.79016 


.62660 


•77934 


.64011 


.76828 


12 


49 


•58519 


.81089 


.59926 


80056 


.61314 


.78998 


.62683 


.77916 


.64033 


.76810 


11 


5° 


•58543 


.81072 


•59949 


80038 


•61337 


.78980 


.62706 


.77897 


.64056 


.76791 


10 


51 


.58567 


.81055 


.59972 


80021 


.61360 


. 78962 


.62728 


•77879 


.64078 


.76772 


I 


52 


•58590 


.81038 


•59995 


80003 


.61383 


.78944 


.62751 


".77861 


.64100 


.76754 


53 


.58614 


.81021 


.60019 


79986 


.61406 


.78926 


•62774 


.77843 


.64123 


.76735 


7 


54 


.58637 


.81004 


.60042 


79968 


.61429 


.78908 


.62796 


.77824 


.64145 


•76717 


6 


55 


.58661 


.80987 


.60065 


79951 


.61451 


.78891 


.62S19 


. 77806 


.64167 


. 7669S 


5 


56 


.58684 


.80970 


.60089 


79934 


.61474 


.78873 


.62842 


.77788 


.64190 


. 76670 


4 


57 


.58708 


•80953 


.60112 


79916 


.61497 


•78855 


.62864 


•77769 


.64212 


.76661 


3 


58 


.58731 


.80936 


.60135 


79899 


.61520 


•78837 


.628S7 


.77751 


.64234 


.76642 


2 


59 


.58755 


.80919 


.60158 


79881 


•61543 


.78819 


.62909 


•77733 


.64256 


.76623 


1 


60 


•58779 


. 80902 


.60182 


79864 


.61566 


.78801 


.62932 


.77715 


.64279 


.76604 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


54 





53' 


' 


5- 


2° 


5 


<° 


5< 


D° 



Natural Sines and Cosines. 



31 



/ 


40° 


4 


i° 


4 


2° 


43° 


44° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


o 


.64279 


.76604 


.65606 


•7547i 


.66913 


•743*4 


.68200 


•73*35 


. 69466 


•7*934 


60 


I 


.64301 


.76586 


.65628 


•75452 


•66935 


•74295 


.68221 


•73**6 


.69487 


•7*9*4 


59 


2 


•64323 


.76567 


.65650 


•75433 


.66956 


•74276 


.68242 


.73096 


. 69508 


.7*894 


58 


3 


.64346 


.76548 


.65672 


•75414 


.66978 


.74256 


.68264 


•73076 


.69529 


•7*873 


57 


4 


.64368 


.76530 


.65694 


•75395 


.66999 


•74237 


.68285 


•73056 


•69549 


.7*853 


56 


5 


.64390 


.76511 


.65716 


•75375 


.67021 


.74217 


.68306 


•73036 


.69570 


•7*833 


55 


6 


.64412 


.76492 


•65738 


•75356 


•67043 


.74198 


.68327 


.73016 


.69591 


•7*8*3 


54 


7 


•64435 


•76473 


•65759 


•75337 


.67064 


.74178 


.68349 


.72996 


.69612 


.71792 


53 


8 


.64457 


•76455 


.65781 


.75318 


.67086 


•74159 


.68370 


.72976 


• 69633 


•7*772 


52 


9 


.64479 


.76436 


.65803 


.75299 


.67107 


•74139 


.68391 


•72957 


.69654 


•7*752 


5* 


IO 


.64501 


.76417 


.65825 


.75280 


.67129 


.74120 


.68412 


•72937 


•69675 


•7*732 


50 


II 


.64524 


.76398 


•65847 


.75261 


.67151 


.74100 


.68434 


•729*7 


.69696 


.71711 


49 


12 


.64546 


.76380 


.65869 


•75241 


.67172 


. 74080 


.68455 


.72897 


.69717 


.71691 


48 


*3 


.64568 


.76361 


.65891 


.75222 


.67194 


.74061 


.68476 


.72877 


•69737 


.71671 


47 


*4 


.64590 


•76342 


•65913 


•75203 


.67215 


.74041 


.68497 


.72857 


•69758 


.71650 


46 


15 


.64612 


.76323 


•65935 


•75184 


.67237 


.74022 


.68518 


•72837 


•69779 


•7*630 


45 


16 


•64635 


.76304 


•65956 


•75165 


.67258 


. 74002 


•68539 


.72817 


. 69800 


.71610 


44 


*7 


•64657 


.76286 


.65978 


75H6 


.67280 


•73983 


.68561 


.72797 


.69821 


•7*59o 


43 


18 


.64679 


.76267 


. 66000 


.75126 


.67301 


•73963 


.68582 


•72777 


.69842 


•7*569 


42 


i9 


.64701 


.76248 


.66022 


.75107 


.67323 


•73944 


.68603 


•72757 


.69862 


•7*549 


4* 


20 


.64723 


.76229 


. 66044 


.75088 


•67344 


•73924 


.68624 


•72737 


.69883 


•7*529 


40 


21 


.64746 


.76210 


.66066 


.75069 


.67366 


.73904 


.68645 


.72717 


. 69904 


•7*508 


39 


22 


.64768 


.76192 


.66088 


.75050 


.67387 


.73885 


.68666 


.72697 


•69925 


.7*488 


38 


2 3 


.64790 


.76173 


.66109 


•75030 


.67409 


.73865 


.68688 


.72677 


.69946 


.71468 


37 


24 


.64812 


•76154 


.66131 


.75011 


.67430 


.73846 


.68709 


.72657 


.69966 


•7*447 


36 


25 


.64834 


•76135 


.66153 


.74992 


.67452 


.73826 


•68730 


.72637 


.69987 


.71427 


35 


26 


.64856 


.76116 


.66175 


•74973 


•67473 


.73806 


.68751 


.72617 


. 70008 


•7*407 


34 


27 


.64878 


.76097 


.66197 


•74953 


•67495 


.73787 


.68772 


.72597 


.70029 


•7*386 


33 


28 


.64901 


.76078 


.66218 


•74934 


.67516 


.73767 


.68793 


.72577 


.70049 


.71366 


32 


29 


• 64923 


.76059 


.66240 


•749 I 5 


.67538 


•73747 


.68814 


.72557 


. 70070 


•7*345 


3* 


3° 


.64945 


.76041 


.66262 


.74896 


.67559 


.73728 


.68835 


•72537 


.70091 


•7*325 


3° 


3 1 


.64967 


.76022 


.66284 


.74876 


.67580 


•737o8 


.68857 


•725*7 


.70112 


•7*305 


29 


3 2 


.64989 


.76003 


.66306 


•74857 


.67602 


.73688 


.68878 


•72497 


.70132 


.71284 


28 


33 


.65011 


.75984 


.66327 


•74838 


.67623 


.73669 


.68899 


•72477 


•70*53 


.71264 


27 


34 


•65033 


•75965 


.66349 


.74818 


.67645 


• 73649 


.68920 


.72457 


.70174 


•7*243 


26 


35 


•65055 


•75946 


.66371 


•74799 


.67666 


.73629 


.68941 


.72437 


•70195 


.7*223 


25 


36 


•65077 


.75927 


.66393 


.74780 


.67688 


.73610 


.68962 


.72417 


.70215 


•7*203 


24 


37 


.65100 


.75908 


.66414 


.74760 


.67709 


•73590 


.68983 


•72397 


.70236 


.71182 


23 


38 


.65122 


.75889 


.66436 


.74741 


.67730 


.73570 


. 69004 


•72377 


.70257 


.71162 


22 


39 


.65144 


•75870 


.66458 


.74722 


•67752 


•73551 


.69025 


•72357 


.70277 


.71141 


21 


40 


.65166 


•75851 


.66480 


•74703 


•67773 


•73531 


.69046 


•72337 


.70298 


.71121 


20 


4i 


.65188 


.75832 


.66501 


.74683 


•67795 


•735II 


. 69067 


•723*7 


•703*9 


.71100 


*9 


42 


.65210 


.75813 


.66523 


.74664 


.67816 


•7349* 


.69088 


.72297 


•70339 


.71080 


18 


43 


.65232 


•75794 


•66545 


•74644 


.67837 


•73472 


.69109 


.72277 


. 70360 


•7*059 


*7 


44 


•65254 


•75775 


.66566 


•74625 


.67859 


•73452 


.69130 


•72257 


.70381 


•7*039 


16 


45 


.65276 


•75756 


.66588 


.74606 


.67880 


•73432 


.69151 


.72236 


. 70401 


.71019 


*5 


46 


.65298 


.75738 


.66610 


.74586 


.67901 


•734 I 3 


.69172 


.72216 


.70422 


.70998 


*4 


47 


.65320 


•75719 


.66632 


•74567 


.67923 


•73393 


.69193 


.72196 


•70443 


.70978 


*3 


48 


.65342 


.75700 


.66653 


•74548 


•67944 


•73373 


.69214 


.72176 


.70463 


•70957 


12 


49 


.65364 


.75680 


.66675 


•74528 


.67965 


•73353 


•69235 


.72156 


.70484 


.70937 


11 i 


50 


.65386 


.75661 


.66697 


•74509 


.67987 


•73333 


.69256 


.72136 


•70505 


.70916 


10 


5i 


.65408 


.75642 


.66718 


.74489 


.68008 


•733H 


•69277 


.72116 


•70525 


. 70896 


9 


52 


•65430 


•75623 


.66740 


.74470 


.68029 


•73294 


.69298 


.72095 


.70546 


•70875 


8 


53 


.65452 


•75604 


.66762 


•74451 


.68051 


•73274 


•693*9 


.72075 


.70567 


•70855 


7 


54 


.65474 


.75585 


.66783 


•74431 


.68072 


•73254 


.69340 


•72055 


•70587 


• 70834 


6 


55 


.65496 


•75566 


.66805 


.74412 


.68093 


•73234 


.69361 


•72035 


. 70608 


.708*3 


5 


56 


.65518 


•75547 


.66827 


•7439 2 


.68115 


•73215 


.69382 


•72015 


.7062S 


•70793 


4 


57 


.65540 


•75528 


.66848 


•74373 


.68136 


•73*95 


.69403 


•7*995 


.70649 


.70772 


3 


58 


.65562 


•75509 


.66870 


•74353 


.68157 


•73*75 


.69424 


•7*974 


.7067c 


.70752 


2 


59 


.65584 


•7549° 


.66891 


•74334 


.68179 


•73*55 


.69445 


•7*954 


. 70690 


•70731 


1 


60 


. 65606 


•75471 


.66013 


•743H 


.68200 


•73*35 


.60466 


•7*934 


.70711 


.70711 





/ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


t 


4< 


f 


4* 


1° 


4 


7° 


4< 


5° 


4, 


-0 



32 



Natural Tangents and Cotangents. 






t 


o° 


1° 


2° 


I 3° 


4° 


1 1 
/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


1 
Tang 


Cotang 


Tang 


Cotang 


o 


.00000 


Innn. 


.01746 


57.2900 


.03492 


28.6363 


.05241 


19.0811 


.06993 


14.3007 


60 


I 


.00029 


3437-75 


•01775 


56.3506 


.03521 


28.3994 


.05270 


i8-9755 


.07022 


14.2411 


59 


2 


.00058 


1718.87 


.01804 


55-4415 


•03550 


28.1664 


.05299 


18.8711 


.07051 


14.1821 


58 


3 


.00087 


1145.92 


•01833 


54-5613 


•03579 


27.9372 


.05328 


18.7678 


.07080 


14-1235 


57 


4 


.00116 


859-43 6 


.01862 


53.7086 


.03609 


27.7117 


•05357 


18.6656 


.07110 


14.0655 


56 


5 


.00145 


687.549 


.01891 


52.8821 


.03638 


27.4899 


•05387 


18.5645 


•07139 


14.0079 


55 


6 


,00175 


572.957 


.01920 


52.0807 


.03667 


27.2715 


.05416 


18.4645 


.07168 


13-9507 


54 


7 


.00204 


491.106 


.01949 


51.3032 


.03696 


27.0566 


•05445 


18.3655 


.07197 


13.8940 


53 


8 


.00233 


429.718 


.01978 


50.5485 


•03725 


26.8450 


.05474 


18.2677 


.07227 


13-8378 


52 


9 


.00262 


381.971 


.02007 


49.8157 


•03754 


26.6367 


•05503 


18.1708 


.07256 


13.7821 


5i 


IO 


.00291 


343-774 


.02036 


49.1039 


.03783 


26.4316 


.05533 


18.0750 


.07285 


13.7267 


50 


ii 


.00320 


312.521 


.02066 


48.4121 


.03812 


26.2296 


.05562 


17.9802 


•07314 


13.6719 


49 


12 


.00349 


2S6.478 


.02095 


47-7395 


.03842 


26.0307 


.05591 


17.8863 


.07344 


13.6174 


48 


*3 


.00378 


264.441 


.02124 


47.0853 


.03871 


25.8348 


.05620 


17.7934 


•07373 


13-5634 


47 


H 


.00407 


245-552 


.02153 


46.4489 


.03900 


25.6418 


.05649 


17.7015 


.07402 


13.5098 


46 


J 5 


.00436 


229.182 


.02182 


45.8294 


.03929 


25-4517 


.05678 


17.6106 


•0743 1 


13.4566 


45 


16 


.00465 


214.858 


.02211 


45.2261 


.03958 


25.2644 


.05708 


170205 


.07461 


13-4039 


44 


i7 


.00495 


202.219 


.02240 


44.6386 


.03987 


25.0798 


•05737 


I7-43H 


.07490 


13-3515 


43 


18 


.00524 


190.984 


.02269 


44.0661 


.04016 


24.8978 


.05766 


I7-3432 


.07519 


13.2996 


42 


i9 


•oo553 


180.932 


.02298 


43.5081 


. 04046 


24.7185 


•05795 


17-2558 


.07548 


13.2480 


4i 


20 


.00582 


171.885 


.02328 


42.9641 


.04075 


24.5418 


.05824 


17.1693 


•07578 


13.1969 


40 


21 


.00611 


163.700 


.02357 


42.4335 


.04104 


24-3675 


.05854 


17.0837 


.07607 


13-1461 


39 


22 


.00640 


156.259 


.02386 


41.9158 


.04133 


24-1957 


.05883 


16.9990 


.07636 


13.0958 


38 


23 


.00669 


149.465 


.02415 


41.4106 


.04162 


24.0263 


.05912 


16.9150 


.07665 


13.0458 


37 


24 


.00698 


143-237 


.02444 


40.9174 


.04191 


23-8593 


.05941 


16.8319 


.07695 


12.9962 


36 


25 


.00727 


1 37-5Q7 


•02473 


40.4358 


.04220 


23.6945 


.05970 


16.7496 


.07724 


12.9469 


35 


26 


.00756 


132.219 


.02502 


39-9655 


.04250 


23-5321 


.05999 


16.6681 


•07753 


12.8981 


34 


27 


.00785 


127.321 


.02531 


39-5059 


.04279 


23-3718 


. 06029 


16.5874 


.07782 


12.8496 


33 


28 


.00815 


122.774 


.02560 


39.0568 


.04308 


23.2137 


.06058 


16.5075 


.07812 


12.8014 


32 


29 


.00844 


118.540 


.02589 


38.6177 


•04337 


23-0577 


.06087 


16.4283 


.07841 


12.7536 


3 1 


3° 


.00873 


114.589 


.02619 


38.1885 


.04366 


22.9038 


.06116 


16.3499 


.07870 


12.7062 


3° 


31 


.00902 


110.892 


.02648 


37.7686 


.04395 


22.7519 


.06145 


16.2722 


.07899 


12.6591 


29 


32 


.00931 


107.426 


.02677 


37-3579 


.04424 


22.6020 


.06175 


16.1952 


.07929 


12.6124 


28 


33 


. 00960 


104. 171 


.02706 


36.9560 


•04454 


22.4541 


,06204 


16.1190 


.07958 


12.5660 


27 


34 


.00989 


101.107 


•02735 


36.5627 


.04483 


22.3081 


.06233 


16.0435 


.07987 


12.5199 


26 


35 


.01018 


98.2179 


.02764 


36.1776 


.04512 


22. 1640 


.06262 


15-9687 


.08017 


12.4742 


25 


36 


.01047 


95-4895 


.02793 


35 . 8006 


.04541 


22.0217 


.06291 


I5-8945 


.08046 


12.4288 


24 


37 


.01076 


92.9085 


.02822 


35-43I3 


.04570 


21.8813 


.06321 


15.8211 


.08075 


12.3838 


23 


38 


.01105 


90-4633 


.02851 


35-0695 


.04599 


21.7426 


.06350 


15.7483 


.08104 


12.3390 


22 


39 


.01135 


88.1436 


.02881 


34-7I5I 


.04628 


21 .6056 


.06379 


15.6762 


.0S134 


12.2946 


21 


40 


.01164 


85.9398 


.02910 


34.3678 


.04658 


21.4704 


.06408 


15.6048 


.08163 


12.2505 


20 


41 


.01193 


83.8435 


.02939 


34.0273 


.04687 


21.3369 


.06437 


I5.5340 


.08192 


12.2067 


19 


42 


.01222 


81.8470 


.02968 


33.6935 


.04716 


21.2049 


.06467 


15.4638 


.08221 


12.1632 


18 


43 


.01251 


79-9434 


.02997 


33.3662 


•04745 


21.0747 


.06496 


15-3943 


.08251 


12.1201 


17 


44 


.01280 


78.1263 


.03026 


33-0452 


.04774 


20 . 9460 


.06525 


15-3254 


.08280 


12.0772 


16 


45 


•01309 


76.3900 


•03055 


32.7303 


.04803 


20.8188 


.06554 


15.2571 


.0S309 


12.0346 


15 


46 


.01338 


74.7292 


.03084 


32.4213 


.04833 


20.6932 


.06584 


15.1893 


•08339 


11.9923 


14 


47 


,01367 


73-i39o 


.03114 


32.1181 


.04862 


20.5691 


.06613 


15.1222 


.08368 


n.9504 


13 


48 


.01396 


71.6151 


.03143 


31.8205 


.04891 


20.4465 


.06642 


15.0557 


•08397 


11.9087 


12 


49 


.01425 


7o-i533 


.03172 


31.5284 


.04920 


20.3253 


.06671 


14.9898 


.08427 


11.S673 


n 


50 


•01455 


68.7501 


.03201 


31.2416 


.04949 


20.2056 


.06700 


14.9244 


.08456 


11.8262 


10 


5i 


.01484 


67.4019 


.03230 


3° -9599 


.04978 


20.0872 


.06730 


14.8596 


08485 


n.7853 


9 


52 


.01513 


66.1055 


.03259 


30.6833 


.05007 


19.9702 


•06759 . 


14-7954 


.08514 


11.7448 


8 


53 


.01542 


64.8580 


.03288 


30.4116 


•05037 


19.8546 


.06788 


14.7317 


.08544 


n.7045 


7 


54 


.01571 


63.6567 


•03317 


30.1446 


.05066 


19.7403 


.06817 


14.6685 


.08573 


n.6645 


6 


55 


.0160a 


62.4992 


.03346 


29.8823 


•05095 


19.6273 


.06847 


14.6059 


.08602 


n.6248 


5 


56 


.01629 


61.3829 


.03376 


29.6245 


.05124 


19-5156 


.06876 


14.5438 


.08632 


".5853 


4 


57 


.01658 


60.3058 


.03405 


29.3711 


.05153 


19.4051 


.06905 


14.4823 


.08661 


n. 5461 


3 


58 


.01687 


59.2659 


.03434 


29. 1220 


.05182 


19.2959 


.06934 


14.4212 


.08690 


11.5072 


2 


59 


.01716 


58.2612 


•03463 


28.8771 


.05212 


19.1879 


.06963 


14.3607 


.08720 


11.4685 


1 


60 


.01746 


57.2000 


.03402 


28.6363 


.05241 


10.0811 


.06993 


14.3007 


.08749 


n.4301 





' 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


1 


89 


88° 


87 


St 





85 






Natural Tangents and Cotangents. 



33 



/ 


5° 


6° 


7° 


8° 


9° 


-1 

; 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


.08749 


1 1. 4301 


.10510 


9.51436 


.12278 


8.14435 


• 14054 


7-H537 


.15838 


6.31375 


60 


I 


.08778 


11. 3919 


. 10540 


9 


.48781 


.12308 


8.12481 


. 14084 


7. 10038 


.15868 


6.30189 


59 


2 


.08807 


H-3540 


.10569 


9 


.46141 


.12338 


8.10536 


.14113 


7.08546 


.15898 


6.29007 


58 


3 


.08837 


11. 3163 


.10599 


9 


.43515 


.12367 


8.08600 


•14143 


7.07059 


-15928 


6.27829 


57 


4 


.06866 


11.2789 


.10628 


9 


. 40904 


.12397 


8.06674 


.14173 


7-05579 


.15958 


6.26655 


56 


5 


.08895 


11. 2417 


.10657 


9 


•38307 


.12426 


8.047^6 


. 14202 


7.04105 


.15988 


6.25486 


55 


6 


.08925 


11.2048 


.10687 


9 


•35724 


.12456 


8.02848 


.14232 


7.02637 


.16017 


6.24321 


54 


7 


.08954 


11.1681 


.10716 


9 


•33155 


.12485 


8.00948 


.14262 


7.01174 


.16047 


6.23160 


53 


8 


.08983 


11. 1316 


.10746 


9 


• 30599 


•12515 


7.99058 


.14291 


6.99718 


.16077 


6.22003 


52 


9 


.09013 


11.0954 


•10775 


9 


.28058 


.12544 


7.97176 


.14321 


6.98268 


.16107 


6.20851 


5i 


IO 


.09042 


n.0594 


.10805 


9 


.25530 


.12574 


7-95302 


•1435 1 


6.96823 


•16137 


6.19703 


50 


n 


.09071 


11.0237 


.10834 


9 


.23016 


.12603 


7-93438 


.14381 


6.95385 


.16167 


6.18559 


49 


12 


.09101 


10.9882 


.10863 


9 


.20516 


.12633 


7.91582 


.14410 


6-93952 


.16196 


6.17419 


48 


13 


.09130 


10.9529 


.10893 


9 


.18028 


.12662 


7-89734 


.14440 


6.92525 


.16226 


6.16283 


47 


14 


.09159 


10.9178 


.10922 


9 


•15554 


.12692 


7.87895 


.14470 


6.91 104 


.16256 


6.15151 


46 


15 


.09189 


10.8829 


.10952 


9 


13093 


.12722 


7 . 86064 


.14499 


6.89688 


.16286 


6.14023 


45 


16 


.09218 


10.8483 


.10981 


9 


10646 


.12751 


7.84242 


.14529 


6.88278 


.16316 


6.12899 


44 


17 


.09247 


10.8139 


. IIOII 


9 


0821 1 


.12781 


7.82428 


•14559 


6.86874 


•16346 


6.11779 


43 


18 


.09277 


10.7797 


. I 1040 


9 


05789 


.12810 


7.80622 


.14588 


6.85475 


•16376 


6.10664 


42 


19 


.09306 


10.7457 


.11070 


9 


•03379 


.12840 


7.78825 


.14618 


6.84082 


•16405 


6.09552 


4i 


20 


•09335 


10.7119 


.11099 


9 


00983 


.12869 


7-77035 


.14648 


6.82694 


.16435 


6.08444 


40 


21 


.o93 6 5 


10.6783 


.11128 


8 


98598 


.12899 


7-75254 


. 14678 


6.81812 


.16465 


6.07340 


39 


22 


.09394 


10.6450 


.11158 


8 


96227 


.12929 


7.73480 


• 14707 


6.79936 


.16495 


6.06240 


38 


23 


.09423 


10.6118 


.11187 


S 


93867 


.12958 


7-71715 


•14737 


6.78564 


.16525 


6.05143 


37 


24 


•09453 


10.5789 


.11217 


S 


91520 


.12988 


7-69957 


.14767 


6.77199 


•16555 


6.04051 


36 


25 


.09482 


10.5462 


.11246 


8 


89185 


.13017 


7.68208 


.14796 


6.75838 


.16585 


6.02962 


35 


26 


.09511 


10.5136 


. 11276 


8 


86862 


.i3 47 


7 . 66466 


.14826 


6.74483 


.16615 


6.01878 


34 


27 


.09541 


10.4813 


.11305 


8 


84551 


.13076 


7.64732 


.14856 


6-73 J 33 


.16645 


6.00797 


33 


28 


.09570 


10.4491 


•I 1335 


8 


82252 


.13106 


7 . 63005 


.14886 


6.71789 


.16674 


5.99720 


32 


29 


. 09600 


10.4172 


.11364 


8 


79964 


.13136 


7.61287 


•14915 


6 . 70450 


.16704 


5.98646 


31 


30 


.09629 


10.3854 


.11394 


8 


77689 


.13165 


7-59575 


• 14945 


6.69116 


•16734 


5-97576 


30 


3i 


.09658 


10.3538 


.11423 


8 


75425 


•13195 


7.57872 


•14975 


6.67787 


.16764 


5.96510 


29 


32 


.09688 


10.3224 


.11452 


8 


73172 


.13224 


7.56176 


.15005 


6.66463 


.16794 


5-95448 


28 


33 


.09717 


10.2913 


.11482 


8 


7093 ! 


•i3 2 54 


7-54487 


.15034 


6.65144 


.16824 


5-9439Q 


*7 


34 


.09746 


10.2602 


.11511 


8 


687OI 


.13284 


7.52806 


.15064 


6.63831 


.16854 


5-93335 


26 


35 


.09776 


10.2294 


.11541 


8 


66482 


•13313 


7-51132 


.15094 


6.62523 


.16884 


5.92283 


25 


36 


.09805 


10.1988 


.11570 


8 


64275 


•13343 


7-49465 


.15124 


6.61219 


.16914 


5.91236 


24 


37 


.09834 


10.1683 


.11600 


8 


62078 


•13372 


7.47806 


•15153 


6.59921 


. 16944 


5.90191 


23 


38 


.09864 


10.1381 


.11629 


8 


59893 


•13402 


7.46154 


.15183 


6.58627 


.16974 


5.89151 


22 


39 


.09893 


10.1080 


.11659 


8 


57718 


•13432 


7-44509 


•15213 


6-57339 


.17004 


5.88114 


21 


40 


.09923 


10.0780 


.11688 


8 


55555 


.13461 


7.42871 


•15243 


6.56055 


• I 7°33 


5.87080 


20 


4i 


.09952 


10.0483 


.11718 


8 


53402 


.13491 


7.41240 


.15272 


6-54777 


.17063 


5.86051 


19 


42 


.09981 


10.0187 


.11747 


8 


51259 


•13521 


7.39616 


.15302 


6.53503 


.17093 


5-85024 


18 


43 


.1001 1 


9.98931 


.11777 


8 


49128 


•13550 


7-37999 


.15332 


6.52234 


.17123 


5.84001 


17 


44 


. 10040 


9.96007 


.11806 


8 


47007 


.13580 


7-36389 


.15362 


6.50970 


•17153 


5.82982 


16 


45 


.10069 


9.93101 


.11836 


8 


44896 


.13609 


7.34786 


.15391 


6.49710 


.17183 


5.81966 


15 


46 


.10099 


9.90211 


.11865 


8 


42795 


.13639 


7-33I90 


.15421 


6.48456 


.17213 


5-80953 


14 


47 


.10128 


9-87338 


.11895 


8 


40705 


.13669 


7.31600 


•15451 


6.47206 


•17243 


5-79944 


13 


48 


.10158 


9.84482 


.11924 


8 


38625 


.13698 


7.30018 


.15481 


6.45961 


•17273 


5-78938 


12 


49 


.10187 


9.81641 


• I 1954 


8 


36555 


.13728 


7.28442 


.15511 


6.44720 


•17303 


5-77936 


11 


50 


.10216 


9.78817 


.11983 


8 


34496 


.13758 


7.26873 


.15540 


6.43484 


•17333 


5-76937 


10 


51 


.10246 


9 . 76009 


.12013 


8 


32446 


.13787 


7.25310 


•15570 


6.42253 


•17363 


5-75941 


9 


52 


. 10275 


9.73217 


.12042 


8 


30406 


.13817 


7-23754 


.15600 


6.41026 


•17393 


5-74949 


8 


53 


• 10305 


9.70441 


.12072 


8 


28376 


.13846 


7.22204 


.15630 


6.39804 


•17423 


5-7396o 


7 


54 


•10334 


9.67680 


.12101 


8 


26355 


.13876 


7.20661 


.15660 


6.38587 


•17453 


5-72974 


6 


55 


• 10363 


9-64935 


.12131 


8 


24345 


.13906 


7.19125 


.15689 


6-37374 


.17483 


5.71992 


5 


56 


.10393 


9.62205 


.12160 


8 


22344 


•13935 


7-17594 


•15719 


6.36165 


.17513 


5-7ioi3 


4 


57 


.10422 


9.59490 


.12190 


8 


20352 


.13965 


7. 16071 


.15749 


6.34961 


•17543 


5 • 70037 


3 


58 


.10452 


9.56791 


.12219 


8 


18370 


.13995 


7-14553 


•15779 


6.33761 


•17573 


5 • 69064 


2 


59 


.10481 


9.54106 


.12249 


8 


16398 


.14024 


7.13042 


.15809 


6.32566 


.17603 


5.6S094 


1 


60 
/ 


. 10510 


q. 51436 


.12278 


8 


14435 


.14054 


7."537 


.15838 


6.31375 


•17633 


5-67128 





Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


' 


84° 


83° 


82° 


Si° 


8o° 



34 



Natural Tangents and Cotangents. 



1 


IO° 


1 »• 


12° 


13° 


14° 


f 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


•17633 


5.67128 


• 19438 


5-14455 


.21256 


4.70463 


.23087 


4-33148 


•24933 


4.01078 


~6o~ 


I 


.17663 


5.66165 


.19468 


5-13658 


.21286 


4 


69791 


.23117 


4-32573 


.24964 


4.00582 


59 


2 


.17693 


5.65205 


.19498 


5.12862 


.21316 


4 


69121 


.23148 


4.32001 


.24995 


4.00086 


58 


3 


•17723 


5.64248 


.19529 


5.12069 


.21347 


4 


68452 


•23179 


4.31430 


.25026 


3-99592 


57 


4 


•17753 


5-63295 


•19559 


5.11279 


•21377 


4 


67786 


.23209 


4 . 30860 


.25056 


3.99099 


56 


5 


.17783 


5-62344 


.19589 


5 . 10490 


.21408 


4 


67121 


.23240 


4.30291 


.25087 


3.98607 


55 


6 


.17813 


5-61397 


. 19619 


5.09704 


.21438 


4 


66458 


.23271 


4.29724 


.25118 


3-98117 


54 


7 


.17843 


5.60452 


.19649 


5 .08921 


.21469 


4 


65797 


.23301 


4.29159 


.25149 


3.97627 


53 


8 


.17873 


5-595I 1 


.19680 


5.08139 


•21499 


4 


65138 


•23332 


4.28595 


.25180 


3-97139 


52 


9 


•i79°3 


5-58573 


.19710 


5.07360 


•21529 


4 


64480 


•23363 


4.28032 


.25211 


3.96651 


5i 


IO 


•17933 


5-57638 


.19740 


5.06584 


.21560 


4 


63825 


.23393 


4.27471 


.25242 


3.96165 


50 


ii 


.17963 


5.56706 


.19770 


5.05809 


.21590 


4 


63171 


.23424 


4.26911 


.25273 


3.95680 


49 


12 


•17993 


5-55777 


.19801 


5-05037 


.21621 


4 


62518 


•23455 


4.26352 


.25304 


3.95196 


48 


13 


.18023 


5-5485I 


.19831 


5.04267 


.21651 


4 


61868 


•23485 


4-25795 


•25335 


3-947I3 


47 


J 4 


.18053 


5-53927 


.19861 


5-03499 


.21682 


4 


61219 


•23516 


4-25239 


.25366 


3-94232 


46 


15 


.18083 


5 •530°7 


.19891 


5-02734 


.21712 


4 


60572 


■23547 


4.24685 


•25397 


3-93751 


45 


16 


.18113 


5.52090 


.19921 


5-OI97I 


•21743 


4 


59927 


•23578 


4.24132 


.25428 


3-93271 


44 


17 


.18143 


5-51176 


• 19952 


5.01210 


•21773 


4 


59283 


.23608 


4.23580 


•25459 


3-92793 


43 


18 


.18173 


5.50264 


.19982 


5.00451 


.21804 


4 


58641 


•23639 


4.23030 


.25490 


3.92316 


42 


19 


.18203 


5-49356 


.20012 


4.99695 


.21834 


4 


58001 


.23670 


4.22481 


.25521 


3-91839 


4i 


20 


•18233 


5.48451 


.20042 


4.98940 


.21864 


4 


57363 


.23700 


4.21933 


•25552 


3-91364 


40 


21 


.18263 


5.47548 


.20073 


4.98188 


.21895 


4 


56726 


•23731 


4.21387 


•25583 


3 . 90890 


39 


22 


.18293 


5.46648 


.20103 


4-97438 


.21925 


4 


56091 


.23762 


4.20842 


.25614 


3.90417 


38 


23 


.18323 


5-45751 


.20133 


4.96690 


.21956 


4 


55453 


•23793 


4.20298 


.25645 


3-89945 


37 


24 


.18353 


5.44857 


.20164 


4-95945 


.21986 


4 


54826 


.23823 


4.19756 


.25676 


3.89474 


36 


25 


.18384 


5-43966 


.20194 


4.95201 


.22017 


4 


54196 


•23854 


4.19215 


•25707 


3 . 89004 


35 


26 


.18414 


5 -43°77 


.20224 


4.94460 


.22047 


4 


53568 


.23885 


4.1S675 


.25738 


3-88536 


34 


27 


.18444 


5.42192 


.20254 


4-93721 


.22078 


4 


52941 


.23916 


4-18137 


.25769 


3 . 88068 


33 


28 


.18474 


5- 4*309 


.20285 


4.92984 


.22108 


4 


52316 


.23946 


4 . 1 7600 


.25800 


3.87601 


32 


29 


.18504 


5.40429 


.20315 


4.92249 


.22139 


4 


51693 


•23977 


4.17064 


.25831 


3-87136 


31 


30 


•18534 


5-39552 


•20345 


4-9i5i6 


.22169 


4 


51071 


.24008 


4.16530 


.25862 


3.86671 


30 


3i 


.18564 


5-38677 


.20376 


4.90785 


.22200 


4 


50451 


•24039 


4-15997 


.25893 


3.86208 


29 


32 


.18594 


5.37805 


. 20406 


4.90056 


.22231 


4 


49832 


.24069 


4.15465 


.25924 


3-85745 


28 


33 


.18624 


5-36936 


.20436 


4.89330 


.22261 


4 


49215 


.24100 


4.14934 


•25955 


3.85284 


27 


34 


.18654 


5 . 36070 


. 20466 


4.88605 


.22292 


4 


48600 


.24131 


4 • 14405 


.25986 


3.84824 


26 


35 


.18684 


5.35206 


.20497 


4.S7882 


.22322 


4 


47986 


.24162 


4-13877 


.26017 


3.84364 


25 


36 


.18714 


5-34345 


.20527 


4.87162 


•22353 


4 


47374 


•24193 


4.i335o 


. 26048 


3.83906 


24 


37 


•18745 


5-33487 


•20557 


4.86444 


.22383 


4 


46764 


.24223 


4.12825 


.26079 


3-83449 


23 


38 


.18775 


5.32631 


.20588 


4-85727 


.22414 


4 


46i55 


.24254 


4.12301 


.26110 


3.82992 


22 


39 


.18805 


5-3I778 


.20618 


4.85013 


.22444 


4 


45548 


.24285 


4-11778 


.26141 


3-82537 


21 


40 


•18835 


5.30928 


.20648 


4 . 84300 


•22475 


4 


44942 


.24316 


4.11256 


.26172 


3.82083 


20 


4i 


.18865 


5.30080 


.20679 


4-83590 


•22505 


4 


44338 


•24347 


4.10736 


.26203 


3.81630 


19 


42 


.18895 


5-29235 


. 20709 


4.82882 


.22536 


4 


43735 


•24377 


4.10216 


.26235 


3-81177 


18 


43 


.18925 


5-28393 


.20739 


4.82175 


.22567 


4 


43134 


.24408 


4.09699 


.26266 


3.80726 


17 


44 


.18955 


5-27553 


.20770 


4.81471 


•22597 


4 


42534 


•24439 


4.09182 


.26297 


3.80276 


16 


45 


.18986 


5.26715 


. 20800 


4.80769 


.22628 


4 


41936 


.24470 


4.08666 


.26328 


3.79827 


15 


46 


.19016 


5.25880 


.20830 


4.80068 


.22658 


4 


41340 


.24501 


4.08152 


•26359 


3-79378 


14 


47 


.19046 


5.25048 


.20861 


4-79370 


.22689 


4 


40745 


•24532 


4.07639 


. 26390 


3-78931 


13 


48 


.19076 


5.24218 


.20891 


4.78673 


.22719 


4 


40152 


.24562 


4.07127 


.26421 


3.78485 


12 


49 


.19106 


5-2339I 


. 2092 1 


4.77978 


.22750 


4 


3956o 


•24593 


4.06616 


. 26452 


3 . 78040 


11 


50 


.19136 


5.22566 


.20952 


4.77286 


.22781 


4 


38969 


.24624 


4.06107 


•26483 


3-77595 


10 


5i 


.19166 


5.21744 


.20982 


4-76595 


.22811 


4 


38381 


.24655 


4-05599 


.26515 


3-77I52 


9 


52 


.19197 


5.20925 


.21013 


4.75906 


.22842 


4 


37793 


.24686 


4.05092 


.26546 


3.76709 


8 


53 


.19227 


5.20107 


.21043 


4-752I9 


.22872 


4 


37207 


•24717 


4.04586 


.26577 


3.76268 


7 


54 


•19257 


5.19293 


.21073 


4-74534 


.22903 


4 


36623 


•24747 


4.04081 


.26608 


3-7582S 


6 


55 


.19287 


5.18480 


.21104 


4-73 8 5i 


•22934 


4 


36040 


.24778 


4-03578 


.26639 


3-7538S 


5 


56 


•19317 


5-17671 


.21134 


4-73i7o 


.22964 


4 


35459 


. 24809 


4.03076 


.26670 


3 • 7495° 


4 


57 


•19347 


5.16863 


.21164 


4.72490 


.22995 


4 


34879 


.24840 


4.02574 


.26701 


3-74512 


3 


58 


.19378 


5.16058 


.21195 


4-7i8i3 


.23026 


4 


34300 


.24871 


4.02074 


.26733 


3-74075 


2 


59 


.19408 


5-15256 


.21225 


4-7II37 


.23056 


4 


33723 


.24902 


4.01576 


.26764 


3-73640 


1 


60 


.19438 


5-14455 


.21256 


4.70463 


•23087 


4 


33148 


•24933 


4.01078 


.26795 


3-73205 




1 


t 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


79° 


78° 


77° 


76 


75° 



Natural Tangents and Cotangents. 



35 



1 


15° 


16 


17° 


18° 


19° 


f 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


.26795 


3-73205 


.28675 


3.48741 


•30573 


3.27085 


.32492 


3.07768 


•34433 


2.90421 


60 


I 


.26826 


3.72771 


.28706 


3-48359 


.30605 


3-26745 


•32524 


3.07464 


•34465 


2.90147 


59 


2 


.26857 


3-7 2 338 


.28738 


3-47977 


•30637 


3.26406 


•32556 


3.07160 


.34498 


2.89873 


58 


3 


.26888 


3.71907 


.28769 


3-47596 


. 30669 


3.26067 


.32588 


3.06857 


•34530 


2 . 89600 


57 


4 


.26920 


3-7*476 


.28800 


3.47216 


.30700 


3-25729 


.32621 


3-06554 


•34563 


2.89327 


56 


5 


•26951 


3.71046 


.28832 


3-46837 


•30732 


3-25392 


•32653 


3.06252 


•34596 


2.89055 


55 


6 


.26982 


3.70616 


.28864 


3.46458 


•30764 


3-25055 


.32685 


3-05950 


•34628 


2.88783 


54 


7 


.27013 


3.70188 


.28895 


3 . 46080 


•30796 


3.24719 


•327*7 


3-05649 


.34661 


2.88511 


53 


8 


.27044 


3.69761 


.28927 


3-45703 


.30828 


3-24383 


•32749 


3-05349 


•34693 


2.88240 


52 


9 


.27076 


3-69335 


.28958 


3-45327 


.30860 


3.24049 


.32782 


3.05049 


.34726 


2.87970 


5* 


IO 


.27107 


3.68909 


.28990 


3-44951 


.30891 


3-237*4 


.32814 


3-04749 


•34758 


2.87700 


50 


ii 


.27138 


3.68485 


.29021 


3-44576 


.30923 


3-23381 


.32846 


3.04450 


•3479* 


2.87430 


49 


12 


.27169 


3.68061 


•29053 


3.44202 


•30955 


3.23048 


.32878 


3-04*52 


.34824 


2.87161 


48 


13 


.27201 


3.67638 


. 29084 


3.43829 


.30987 


3.22715 


•329*1 


3-03854 


.34856 


2.86892 


47 


14 


.27232 


3.67217 


.29116 


3-43456 


•3*o*9 


3.22384 


.32943 


3-03556 


.34889 


2.86624 


46 


15 


.27263 


3 . 66796 


.29147 


3.43084 


•3*051 


3.22053 


.32975 


3.03260 


.34922 


2.86356 


45 


16 


.27294 


3.66376 


.29179 


3-427*3 


•3*083 


3.21722 


•33007 


3.02963 


•34954 


2.86089 


44 


17 


.27326 


3-65957 


.29210 


3-42343 


•3***5 


3.21392 


.33040 


3.02667 


•34987 


2.85822 


43 


18 


•27357 


3-65538 


.29242 


3-4*973 


•3**47 


3.21063 


•33072 


3.02372 


•35020 


2 -85555 


42 


19 


.27388 


3-65121 


.29274 


3.41604 


.31178 


3.20734 


•33*04 


3.02077 


•35052 


2.85289 


4* 


20 


.27419 


3-64705 


•29305 


3.41236 


.31210 


3 . 20406 


•33*36 


3-01783 


•35085 


2.85023 


40 


21 


•27451 


3.64289 


•29337 


3 . 40869 


.31242 


3.20079 


•33*69 


3.01489 


•35**8 


2.84758 


39 


22 


.27482 


3-63874 


.29368 


3.40502 


.3*274 


3-19752 


.33201 


3.01196 


.35*50 


2.84494 


38 


23 


•27513 


3.63461 


.29400 


3.40136 


•3*306 


3.19426 


•33233 


3.00903 


•35*83 


2.84229 


37 


24 


.27545 


3.63048 


.29432 


3-3977* 


.3*338 


3.19100 


.33266 


3. 0061 1 


•35216 


2.83965 


36 


25 


•27576 


3.62636 


•29463 


3-394o6 


•3*370 


3-18775 


•33298 


3.00319 


•35248 


2.83702 


35 


26 


.27607 


3.62224 


.29495 


3-39042 


.3*402 


3-1845* 


•33330 


3.00028 


.35281 


2.83439 


34 


27 


.27638 


3.61814 


.29526 


3-38679 


•3*434 


3-18127 


•33363 


2.99738 


•353*4 


2.83176 


33 


28 


.27670 


3.61405 


•29558 


3-383*7 


.3*466 


3.17804 


•33395 


2.99447 


•35346 


2.82914 


32 


29 


.27701 


3 . 60996 


.29590 


3-37955 


.31498 


3-17481 


•33427 


2.99158 


•35379 


2.82653 


31 


3° 


.27732 


3.60588 


.29621 


3-37594 


•3*53° 


3-*7*59 


•3346o 


2.98868 


•354*2 


2.82391 


30 


31 


.27764 


3.60181 


•29653 


3-37234 


.3*562 


3.16838 


•33492 


2.98580 


•35445 


2.82130 


29 


32 


•27795 


3-59775 


.29685 


3-36875 


•3*594 


3-i65*7 


•33524 


2.98292 


•35477 


2.81870 


28 


33 


.27826 


3- 59370 


.29716 


3-365*6 


.31626 


3.16197 


•33557 


2.98004 


•355*o 


2.81610 


27 


34 


.27S58 


3.58966 


.29748 


3.36*58 


.3*658 


3-I5877 


•33589 


2.97717 


•35543 


2.81350 


26 


35 


.27889 


3.58562 


.20780 


3 • 358oo 


.3*690 


3.15558 


•33621 


2.97430 


•35576 


2.81091 


25 


36 


.27921 


3.58160 


.29811 


3-35443 


.3*722 


3.15240 


•33654 


2.97144 


.35608 


2.80833 


24 


37 


.27952 


3-57758 


•29843 


3-35o87 


•3*754 


3.14922 


.33686 


2.96858 


.35641 


2.80574 


23 


38 


.279S3 


3-57357 


•29875, 


3-34732 


.3*786 


3.14605 


•337*8 


2.96573 


•35674 


2.80316 


22 


39 


.28015 


3-56957 


.29906 


3-34377 


.31818 


3.14288 


•3375* 


2.96288 


•35707 


2 . 80059 


21 


40 


.28046 


3-56557 


.29938 


3-34023 


.3*850 


3-I3972 


•33783 


2 . 96004 


•3574o 


2 . 79802 


20 


4i 


.28077 


3-56i59 


.29970 


3-33670 


.31882 


3.13656 


.33816 


2.95721 


•35772 


2-79545 


*9 


42 


.28109 


3-5576i 


.30001 


3-333*7 


•3*9*4 


3-*334* 


.33848 


2-95437 


•35805 


2.79289 


18 


43 


.28140 


3-55364 


.30033 


3-32965 


.31946 


3.13027 


.33881 


=■95*55 


.35838 


2 . 79033 


*7 


44 


.28172 


3-54968 


.30065 


3.32614 


.3*978 


3-* 2 7*3 


•339*3 


2.94872 


.3587* 


2.78778 


16 


45 


.28203 


3-54573 


.30097 


3.32264 


.32010 


3. 12400 


•33945 


2-9459* 


.35904 


2 78523 


*5 


46 


.28234 


3 -54*79 


.30128 


3-3*9*4 


.32042 


3.12087 


•33978 


2.94309 


•35937 


2.78269 


*4 


47 


.28266 


3-53785 


.30160 


3-3*565 


.32074 


3-1*775 


.34010 


2.94028 


•35969 


2.78014 


*3 


48 


.28297 


3-53393 


.30192 


3.31216 


.32106 


3-1*464 


•34043 


2.93748 


. 36002 


2.77761 


12 


49 


.28329 


3-53 001 


.30224 


3.30868 


•32139 


3-11*53 


•34075 


2.93468 


•36035 


2.77507 


11 


50 


.28360 


3.52609 


.30255 


3.30521 


.32171 


3.10842 


.34108 


2.93189 


.36068 


2.77254 


10 


5i 


.28391 


3.52219 


.30287 


3-30I74 


.32203 


3-i o 532 


.34140 


2.92910 


.36101 


2.77002 


9 


52 


.28423 


3.51829 


•30319 


3.29829 


•32235 


3.10223 


•34*73 


2 .92632 


.36134 


2.76750 


8 


53 


.28454 


3.5I44* 


•30351 


3.29483 


.32267 


3.09914 


•34205 


2-92354 


.36*67 


2.76498 


7 


54 


.28486 


3-5 io 53 


.30382 


3-29*39 


.32299 


3.09606 


•34238 


2.92076 


.36199 


2.76247 


6 


55 


.28517 


3.50666 


.30414 


3-28795 


•32331 


3.09298 


•34270 


2.91799 


.36232 


2.75996 


5 


56 


.28549 


3.50279 


.30446 


3*28452 


•32363 


3.08991 


•34303 


2.9*523 


.36265 


2.75746 


4 


57 


.23580 


3.49894 


.30478 


3.28109 


.32396 


3.08685 


•34335 


2.91246 


.36208 


2.75496 


3 


58 


.23612 


3.49509 


•30509 


3.27767 


.32428 


3-08379 


•34368 


2.90971 


.3633* 


2.75246 


2 


59 


.28643 


3.49125 


•3054* 


3.27426 


.32460 


3.08073 


.34400 


2.90696 


•36364 


2.74997 


1 


60 


.28675 


3.48741 


•30573 


3.27085 


.32492 


3.07768 


•34433 


2.Q0421 


• 36397 


2.74748 





; 


Cotang 


Tang 


Cotang 


Tang ' 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


• 


74° 


73° 


72° 


7i° 


7°° 



36 



Natural Tangents and Cotangents. 



r 

I- 

1 


20° 


21° 


22° 


"3° 


= 4° 




Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang Cotang 


o 


■y-:-- 


2-7474: 


.3838- 


2.2,5., 


.4:4-5 


-.4-5- r 


.42447 


2-35585 


•44523 


:.--:•-- 


60 


i 


.36430 


2.-44;. 




2.60283 


.4.4.;: 


2.47302 




2-35395 


■ ^y 


2.2442S 


59 


2 


.36463 


2-7425I 


.38453 


2.60057 


.40470 


2.--:. 5 


.42516 


2.35205 


.445:5 


2.24252 


58 


3 


.524-2 


2.74004 




2.52::: 


.4:5:4 


2.--5_ : 


.42551 


'-■ -y 1 


.-- :- 


2.24077 


57 


4 


•:■:-; 


2.73756 


■;:5--' 


2.522:: 


•40538 


2.455i2 


.42555 


2-34825 


.44662 


2 . 23902 


56 


5 


.36562 


2.73509 


•;::55 


2.59381 


•- 5"- 


2.46476 


.42619 


2.34636 


.— ::- 


2 23723 


55 


6 


•3 6 595 


2.73263 


._-S ; E- 


2.51:5: 


.4:5:5 


2 .422-: 


.42654 


2.34447 


•44732 


2-23553 


54 


7 


.: :- 


2.73017 


.-:2: 


2 . = : : - 2 


,_::'4: 


2.46065 


.425;: 


2.3425S 


.44767 


: . 


33 


8 


.]■.": : 


2.72771 


.:55 5 _ 


2.--1-] ; 


.422-4 


2-4=3:: 


.42722 


:.:-::: 


.445:2 


2.23204 


52 


9 


■:" : '-- 


2.-252: 


.:-■- 


- 5:-:- 


.40707 


2-45655 


•42757 


2.352:: 


.4435- 


2.25:5: 


5i 


IO 


.36727 


2 . 7228 : 


.;:-.: 


2.5S26I 


.4074I 


2.45451 


.42791 


-yyy 


.44872 


2.22857 


50 


ii 


•5 ~- : 


2.72036 


.35754 


2. = ::-,: 


•40775 


2.45246 


.42S26 


2-33505 


.44907 


2.22683 


49 


12 


■: \5 


2.71792 


.ni-l- 


2.57815 


...::: 


2.45043 


..21-.Z 


2.33317 


.442-2 


2.22510 


43 


13 


.::£__ 


2.71548 


.-y.i2-. 


2.57593 


.-:54 3 


2.44:5: 


.42894 


2.33130 


•44977 


2.22337 


47 


M 


• = ■' = 5; 


2.71305 


■1^:- 


2.57371 


.40877 


2.44:5: 


•42029 


-•32:43 


.45012 


2.22164 


46 


15 


■5 - 


2.7: :2 


.yiy 


2.57150 


._:;:: 


2-44433 


.422:5 


2.32756 


.45:4- 


2.21992 


45 


16 


■;■ ---= 


2.7:2:; 


.55:2: 


2.5:22: 


.40945 


2.44230 


.-222: 


2.525-: 


.452:2 


2.2:5:: 


44 


17 


■ 3-';5 = 


2.70577 


■3^55 


2.5:'- - 


.40979 


2.44027 


•43032 


2.32383 


.45117 


2.21647 


43 


iS 


• i - : 


2.70335 


■3-?-- 


2-5--:- 


.41013 


2.43825 


.4-2:7 


2.32197 


.45152 


2.21475 


42 


19 


•37024 


2 . " : ; 2 4 


.51:22 


2.5:2:: 


.41047 


2.45:25 


.43101 


2.32012 


.-=:;- 


2 . 21304 


4i 


20 


•37057 


2.69853 


- J - - 5 5 


2.5:24: 


.41081 


2.43422 


•43 -3 6 


: . 3 : 5 ; 5 


.45222 


2.21132 


4: 


21 


•37090 


2 .-';:': 2 


■3;-- r ; 


2 55:;- 


.4i"5 


2.43220 


•43170 


2.31641 


•45257 


2.2:222 


35 


22 


•37123 


;.::-: 


.52:22 


- 55 -.: 


.41149 


2.45::: 


-5-5 


2.31456 


.45222 


2.22-2: 


3 = 


23 


•37*57 


2.::::: 


■3--3- 


- 5 5 5 - 


.41183 


2.42::: 


.45230 


2.31271 


•45327 


2.20619 


37 


24 


■r- r- 


2.68=22 


.52:2: 


2.55170 


.41217 


2.42618 


.43274 


2.5::::' 


.455:2 


2 .2:442 


36 


25 


.37223 


2 . : I 2 : - : 


•39223 


2.5-252 


.41251 


2._24:; 


-y-^ 


2 .5:222 


.4552- 


2.2:2-: 


35 


26 


•37256 


2. :-:_ 


■3«5r 


2-54734 


.41285 


2.42218 


•43343 


2.5:-:: 


•45432 


2.2010S 


34 


27 


.:-_:; 


2.:I:- 5 


■5222: 


2-:-f:: 


•4I3I9 


2.-22:2 


■--y- 


2-30534 


•45467 


2.19938 


33 


23 


.37322 


2 .-:;- - 




2.54209 


•41353 


2.4:2:: 


•43412 


2- 3035 - 


-55:2 


2.19769 


32 


2y 


■37355 


2 . :--:•: 


•39357 


2.54082 


■ -■-:--- 


2.4:22: 


•43447 


2.30167 


■-::}: 


2.10590 


3i 


33 


•:-:- 


2 . . -_:2 


yy- 


2.53865 


.41421 


2.41421 


.43481 


2.22154 


•45573 


2.19430 


3° 


3 1 


•37422 


2.67225 


•39425 


2.53648 


.41455 


2.41223 


.43516 


2.21::: 


.45608 


2.10261 


29 


32 


:'-■ 


2.:::; : 


■s;-5= 


2-53432 


.4:422 


:. 4:225 


■-:::- 


2.20619 


.45643 


:.::::: 


28 


?:- 


•374S8 


2 . 2:7=2 


.52-22 


2.53217 


.41524 


2.42:2- 


.435S5 


2.2:45- 


.45:-: 


2 . :::: : 


27 


:-- 


.37521 


2.22=1; 


■ :-;f;-' 


2.53001 


.4155S 


2 . _ : 5 2 : 


.-5222 


2.20254 


•45713 


2.18755 


25 


55 


■3'::- 


2.:"22i: 


yyy 


2.52-5: 


.4:5:2 


2.4:452 


■ ±y-:- 


;.;::--- 


•-:"-: 


_.:E=3- 


25 


35 


.;- 5 :5 


2 . ::':,:' 


•y-yi 


2.52571 


.4:::: 


2.40235 


.455:; 


2.2:::: 


.45784 


-.18415 


24 


:•" 


.37621 


2.25:2: 


. ---- 21 


- :-- 


.41660 


2 . _ : : 5 : 


•43724 


2.2:-:: 


.45::: 


2.182=1 


23 


33 


■3"-54 


- ■■":3-: 


■3"'-'= 


2.52142 


.-:"24 


2.5::-: 


■-yy 


2.2:=:: 


■-:=:- 


:.:3:54 


22 


39 


■:- - 


-• '-:?-- 


■3-"'- 


2.51929 


.4:-:: 


2.30645 


■ -I'--: 


2.2:5-: 


..5552 


2.17916 


21 


40 


.37720 


2 . v 5 : : 2 


•39727 


2.5I7I5 


.41763 


2.39449 


.43S28 


; . : 5 : 5 - 


.45924 


2.17749 


20 


41 


■37754 


2.64875 


•3976i 


;.=:=:: 


._:-:- 


2.39253 


.4:552 


2.2-2!- 


.452:': 


2.17582 


19 


42 


.:-:- 


2 . :_:_2 


•3;-;= 


2.5:2:2 


.41831 


2.52:5; 


■•^3:5- 


2.27S06 


-:-': 


2.17416 


18 


43 


•:--- : 


2 . 64410 


.52:22 


2.51076 


.4186S 


2.-iy- 


•4393- 


2.27626 


.42:52 


2.1724; 


17 


44 


- 


2.64177 


.52:22 


2.52: 5_ 


.41899 


2.]l2Zl 


._;-;:: 


2.27447 


■-^-5 


2.17083 


16 


45 


.37887 


-■lr-: 


.523.2 


2.50652 


.4:255 


- y--i 


.44001 


2.2-22" 


.46101 


2.16917 


15 


46 


37920 


2.63714 


•39930 


2.50440 


.42:55 


:.:::-: 


.44036 


2.2-::; 


.42:5: 


2. 1675 1 


14 


47 


37953 


2.63483 


•3;;-3 


2.50229 


.42002 


2.5:234 


.44071 


2.2:'::: 


.46171 


2 ■-■;'-: 


13 


4 3 


. 5 - : : : 


2.63252 


.5.222- 


2.5:::; 


.42036 


2.5-3;: 


.44105 


2.22-:: 


.4:':: :' 


2.16420 


12 


49 


. ;:_:: 


2.63021 


._:::: 


2 ._ .: " 


.42 -: 


2.3-52- 


.44140 


2.26552 


.45242 


2.16255 


11 


50 


.38053 


2.-27;: 


.40065 


2.49597 


.42105 


- yy- 


•44175 


2.2:5-4 


•46277 


2 . 222 2- 


10 


51 


.-,:;Sf 


2 . :' 2 5 :' : 


.40098 


2.493S6 


•42139 


2-373" 


.44210 


2.2":;:" 


.46312 


2.I5925 


9 


52 


.3812O 


2.62332 


.4:252 


2.49177 


•42173 


2.5-::: 


.44244 


2.260I8 


.46348 


2.I5760 


8 


53 


■38153 


2 . : 2 : : 5 


.4.:;:: 


2.4:22- 


.42207 


2-y-2: 


.44279 


2.2=54: 


•--5:5 


2-I3?;- 


7 


54 


.5:2:2 


2.61S74 


.-22.2 


2--y 


.42242 


2}-y 


•44314 


2:5 5 


.-:-:: 


2.15432 


6 


55 


.38220 


2 .61646 


•40234 


' 


.42276 


2.36541 




2.25486 


.4:4=4 


2.i=2f; 


5 


:-' 


.38253 


2.61418 


.40267 


2 . _ : 3 _ 


.42310 


2.5:342 


*4_ - 


2.25309 


.4:4:: 


2.15104 


4 


57 


■ y-2 : -'- 


2.61100 


.40301 


2.48132 


.42345 


2.55:5! 


.44418 


2.25132 


■-':2l 


2 . I494O 


3 


58 


.38320 


2.6c 22 5 


•40335 


2.47924 


■42379 


2.5=2:- 


•44453 


2.2-2=2 


■-':' 


2.14777 


2 


59 


•38353 


2.60736 


.4036a 


2.47716 


•42413 


2.35776 


.444I 


2 .24-: : 


. 4 ' 5 ; 5 


2.I4614 


1 




. : : -: 


2 . 60509 | 


.40403 


2.4-=:: 


.4244" 


2 . 5 = = : 5 


.44523 


2.24604 


.4'-:: 


2.144=1 





/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


69= 


68° 


67^ 


66= | 


65= 



Natural Tangents and Cotangents. 



3? 





*5° 


26° 


2 


7° 


2 


3° 


2 


*° 


— 1 

1 
I 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


.46631 


2.14451 


•48773 


2.05030 


•50953 


1. 96261 


•53*7* 


1.88073 


•5543* 


1.80405 


60 


1 I 


.46666 


2.14288 


.48809 


2.04879 


.50989 


1. 96120 


.53208 


1. 87941 


•55469 


1. 80281 


59 


1 
2 


.46702 


2.14125 


.4S845 


2 .04728 


.51026 


1-95979 


-53246 


1.87809 


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1. 80158 


58 


3 


.46737 


2.13963 


.48881 


2.04577 


•5*063 


1.95838 


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1.87677 


•55545 


1 .80034 


57 


4 


.46772 


2.13801 


.48917 


2.04426 


.51099 


1.95698 


•53320 


1.87546 


.55583 


1. 7991 1 


56 


5 


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2.04276 


.51136 


1-95557 


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1. 87415 


.55621 


1.79788 


55 


6 


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2-13477 


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2.04125 


•5 II 73 


1-954*7 


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1.87283 


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1.79665 


54 


7 


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2.13316 


.49026 


2.03975 


.51209 


1.95277 


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1. 87152 


•55697 


1.79542 


53 


8 


.46914 


2.13154 


.49062 


2.03825 


.51246 


1. 95*37 


•53470 


1 .87021 


•55736 


1. 79419 


52 


9 


.46950 


2.12993 


. 49098 


2.03675 


.51283 


1.94997 


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1. 86891 


•55774 


1.79296 


5*' 


IO 


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2.12832 


•49134 


2.03526 


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1.94858 


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1.S6760 


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1. 791 74 


50 


ii 


.47021 


2.12671 


.49170 


2.03376 


.51356 


1. 94718 


.53582 


1.86630 


•55850 


1-7905* 


49 


12 


.47056 


2.12511 


.45206 


2.03227 


•5*393 


1-94579 


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1.86499 


.55888 


1.78929 


48 


*3 


.47092 


2.12350 


.49242 


2.03078 


•5*430 


1.94440 


•53657 


1.86369 


•55926 


1.78807 


47 


14 


.47128 


2.12190 


.49278 


2.02929 


•5*467 


1. 943oi 


•53694 


1.86239 


•55964 


1.78685 


46 


15 


.47163 


2. 12030 


•493*5 


2.02780 


•5*503 


1.94162 


•53732 


1. 86109 


.56003 


1.78563 


45 


16 


.47199 


2.11871 


•49351 


2.02631 


•5*540 


1.94023 


•53769 


1.85979 


.56041 


1. 78441 


44 


17 


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2.11711 


•49387 


2.02483 


•5*577 


1.93885 


•53807 


1.85850 


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1. 78319 


43 


18 


.47270 


2.11552 


.49423 


2.02335 


.5*614 


1.93746 


•53844 


1.85720 


.56117 


1. 78 198 


42 


19 


•473°5 


2.11392 


•49459 


2.02187 


.5*651 


1.93608 


.53882 


1. 85591 


•56*56 


1.78077 


4* 


20 


•47341 


2.11233 


•49495 


2.02039 


.51688 


1.93470 


•53920 


1.85462 


•56i94 


1-77955 


40 


21 


•47377 


2. 1 1075 


•49532 


2.01891 


•5*724 


1-93332 


•53957 


1-85333 


.56232 


1.77834 


39 


22 


.47412 


2 . 10916 


.49568 


2.01743 


.51761 


1. 93*95 


•53995 


1.85204 


.56270 


1-777*3 


38 


23 


.47448 


2.10758 


.49604 


2.01596 


.51798 


I.93057 


•54032 


1-85075 


.56309 


1.77592 


37 


24 


•47483 


2 . 10600 


.49640 


2.01449 


.5*835 


1.92920 


.54070 


1.84946 


.56347 


1. 77471 


36 


25 


•47519 


2. 10442 


.49677 


2.01302 


.5*872 


1 .92782 


•54*o7 


1. 84818 


.56385 


*-7735* 


35 


26 


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2.10284 


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2.01155 


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1.92645 


•54*45 


1.84689 


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1.77230 


34 


27 


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2. 10126 


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2.01008 


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1.92508 


•54*83 


1. 84561 


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1.77110 


33 


28 


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2.09969 


.49786 


2.00862 


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1. 92371 


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1-84433 


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1 . 76990 


32 


29 


.47662 


2. 098 1 1 


.49822 


2.00715 


.52020 


1.92235 


.54258 


1.84305 


•56539 


1.76869 


3* 


3° 


.47698 


2.09654 


.49858 


2.00569 


•52057 


1.92098 


.54296 


1.84177 


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1.76749 


30 


31 


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2 . 09498 


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2.00423 


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1. 91962 


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1 . 84049 


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29 


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1.91826 


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1.83922 


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1. 765 10 


28 


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2.09184 


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2.00131 


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1. 91690 


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1.83794 


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1.76390 


27 


34 


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2.09028 


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1.99986 


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1-9*554 


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1.83667 


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1. 76271 


26 


35 


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2.08872 


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1. 99841 


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1.9*4*8 


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1.83540 


.56769 


1. 7615* 


25 


36 


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2.08716 


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1.99695 


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1. 91282 


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1 .76032 


24 


37 


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i.9955o 


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1.91147 


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1.83286 


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1-759*3 


23 


38 


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1.99406 


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1 .91012 


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1. 83159 


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1-75794 


22 


39 


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2.08250 


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1. 99261 


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1.90876 


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1-83033 


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21 


40 


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2.08094 


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1 .99116 


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1. 90741 


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1.82906 


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I-75556 


20 


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1.98972 


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1.90607 


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1.82780 


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1-75437 


19 


42 


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1.98828 


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1.90472 


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1.82654 


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1-753*9 


18 


43 


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2.07630 


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1.98684 


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* -9°337 


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1.82528 


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1.75200 


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44 


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1.98540 


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1.90203 


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1.82402 


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1.75082 


16 


45 


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1.98396 


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1 . 90069 


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1.82276 


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1.74964 


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1.98253 


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1.98110 


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1. 89801 


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1.74610 


12 


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I-89533 


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II 


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1. 97681 


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1 . 89400 


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1. 81649 


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1-74375 


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9 


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8 


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1 . 89000 


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4 


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1.96685 


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3 


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1 


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1.88073 


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* • 73205 





; 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


; 


6 4 ° 


63° 


6 


2° 


6 


1° 


6 


o° 



38 



.Natural Tangents and Cotangents. 



/ 


3°° 


3i° 


32° 


33° 


34° 


1 

60 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


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1.73205 


.60086 


1.66428 


.62487 


1.60033 1 


.64941 


1.53986 


.67451 


1.48256 


I 


•57774 


1.73089 


.60126 


1. 66318 


.62527 


i-5993o 


.64982 


1.53888 


•67493 


1. 48163 


59 


2 


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1.72973 


.60165 


1 .66209 


.62568 


1.59826 


.65024 


I-5379I 


•67536 


1.48070 


58 


3 


•57851 


1.72857 


. 60205 


1 . 66099 


.62608 


1-59723 


.65065 


1.53693 


•67578 


1.47977 


57 


4 


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1. 72741 


.60245 


1.65990 


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1 . 59620 


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1-53595 


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1.47885 


56 


5 


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1.72625 


.60284 


1. 65881 


.62689 


i-595i7 


.65148 


1-53497 


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1.47792 


55 


6 


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1.72509 


.60324 


1.65772 


.62730 


1. 59414 


.65189 


1.53400 


.67705 


1.47699 


54 


7 


.58007 


1.72393 


.60364 


1.65663 


.62770 


1-593" 


.65231 


1.53302 


•67748 


1.47607 


53 


8 


. 58046 


1.72278 


.60403 


1-65554 


.62811 


1.59208 


.65272 


1-53205 


.67790 


1. 47514 


52 


9 


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1. 72163 


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1.65445 


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1. 59105 


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i-53io7 


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1.47422 


5i 


IO 


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1.72047 


.60483 


1-65337 


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1 . 59002 


•65355 


1. 53010 


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1-4733° 


50 


ii 


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1. 71932 


.60522 


1.65228 


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1.58900 


•65397 


1. 52913 


•67917 


1.47238 


49 


12 


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1.71817 


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1. 65120 


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I-58797 


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1. 52816 


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1. 47146 


48 


13 


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1. 71702 


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1. 6501 1 


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1.58695 


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1. 52719 


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1.47053 


47 


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1. 71588 


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1.64903 


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I-58593 


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1.52622 


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1.46962 


46 


15 


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i-7 x 473 


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1.64705 


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1.58490 


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1-52525 


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1.46870 


45 


l6 


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I.7I358 


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1.64687 


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1.58388 


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1.52429 


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1.46778 


44 


17 


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1. 71244 


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1.64579 


•63177 


1.58286 


.65646 


1.52332 


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1.46686 


43 


18 


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1.71129 


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1. 6447 1 


•63217 


1.5S184 


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1.52235 


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1-46595 


42 


19 


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1. 71015 


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1.64363 


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1.58083 


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4i 


20 


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1.64256 


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1. 57981 


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1.52043 


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1. 4641 1 


40 


21 


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1.70787 


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1. 64148 


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I.57879 


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1.51946 


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1.46320 


39 


22 


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1.70673 


. 60960 


1. 6404 1 


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1-57778 


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1. 51850 


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1.46229 


38 


23 


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1 .70560 


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1-63934 


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1.57676 


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i-5i754 


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1-46137 


37 


24 


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1.70446 


. 6 1 040 


1.63826 


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1-57575 


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1.51658 


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1.46046 


36 


25 


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1.70332 


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1. 63719 


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!• 57474 


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1. 51562 


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1-45955 


35 


26 


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1. 702 19 


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1. 63612 


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1-57372 


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1. 51466 


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1.45864 


34 


27 


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1. 70 106 


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1-63505 


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1-57271 


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1-51370 


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1-45773 


33 


28 


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1.69992 


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1.63398 


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1. 57170 


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1. 51275 


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1 .45682 


32 


29 


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1.69879 


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1.63202 


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1.57069 


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1-51179 


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1-45592 


3i 


3° 


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1.69766 


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1. 63185 


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1.56969 


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1. 51084 


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1-45501 


30 


31 


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1.69653 


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1.63079 


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1.56868 


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1 . 50988 


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1.45410 


29 


32 


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1. 69541 


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1.62972 


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1.56767 


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1.50893 


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1.45320 


28 


33 


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1.69428 


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1.62866 


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1.56667 


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1.50797 


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1.45229 


27 


34 


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1.69316 


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1 .62760 


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1.56566 


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1 .50702 


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L45139 


26 


35 


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1.69203 


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1.62654 


•63912 


1.56466 


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1.50607 


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1.45040 


25 


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1. 6909 1 


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1.62548 


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1.56366 


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1. 50512 


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1.44958 


24 


37 


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1.68979 


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1.62442 


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1.56265 


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1. 50417 


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1.44868 


23 


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1.68866 


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1.62336 


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1. 56165 


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1.50322 


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1.44778 


22 


39 


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1.68754 


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1.62230 


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1.56065 


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1 .50228 


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1.44688 


21 


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1. 62125 


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1.55966 


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20 


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19 


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1.61914 


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1.49944 


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18 


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1.55666 


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1.49849 


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17 


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1. 61 703 


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16 


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1.6159S 


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1.55467 


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1.49661 


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1. 44140 


15 


46 


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1.67974 


.61922 


1.61493 


.64363 


1.55368 


.66860 


1.49566 


.69416 


1 . 44060 


14 


47 


•59573 


1.67863 


.61962 


1. 61388 


.64404 


1.55269 


.66902 


1.49472 


.69459 


1.43970 


13 


4 s 


.59612 


1.67752 


.62003 


1. 61283 


. 64446 


1-55*70 


.66044 


1.49378 


.60502 


1. 43881 


12 


49 


•59651 


1. 67641 


.62043 


1.61179 


.64487 


1. 5507 1 


.66986 


1.49284 


.69545 


1.43792 


11 


50 


.59691 


1.67530 


.62083 


1. 61074 


.64528 


1.54972 


.67028 


1. 49190 


.69588 


I-43703 


10 


51 


•59730 


1. 67419 


.62124 


1 .60970 


.64569 


I-54873 


.67071 


1.49097 


.69631 


1. 43614 


9 


52 


•5977° 


1.67309 


.62164 


1.60865 


.64610 


1-54774 


.67113 


1.49003 


•69675 


1-43525 


8 


53 


.59809 


1. 67198 


.62204 


1. 60761 


.64652 


I-54675 


•67155 


1 48909 


.69718 


I-4343 6 


7 


54 


■59849 


1.67088 


.62245 


1.60657 


.64693 


1-54576 


.67197 


1.48S16 


.69761 


1-43347 


6 


55 


.59888 


1.66978 


.62285 


1.60553 


•64734 


1.54478 


.67239 


1.48722 


. 69804 


1.43258 


5 


56 


.59928 


1.66867 


.62325 


1 . 60449 


•64775 


1-54379 


.67282 


1.48629 


.6r8 47 


1. 43169 


4 


57 


.59967 


1.66757 


.62366 


1.60345 


.64817 


1. 54281 


.67324 


1.48536 


.69891 


T .43080 


3 


58 


.60007 


1.66647 


.62406 


1. 60241 


.64858 


1. 54183 


.67366 


1.48442 


•69934 


I.42992 


2 


59 


.60046 


,.66538 


.62446 


1. 60137 


. 64899 


1-54085 


.67409 


1.48349 


•69977 


I.42903 


1 


60 


.60086 


1.66428 


.62487 


1.60033 


.64041 


1.53986 


•67451 


T. 48256 


.70021 


I.428T5 





/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


5 


9° 


s 


8° 


1 5 


7° 


. 5 


5° 


5 


5° 



Natural Tangents and Cotangents. 



39 



1 


35° 


36° 


37° 


38° 


39° 


/ 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


.70021 


1. 42815 


.72654 


1.37638 


-75355 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


60 


I 


.70064 


1.42726 


.72699 


1-37554 


.75401 


1.32624 


.78175 


1. 27917 


.81027 


1. 23416 


59 


2 


.70107 


1.42638 


•72743 


I-3747Q 


•75447 


1.32544 


.78222 


1. 27841 


.81075 


1-23343 


58 


3 


.70151 


1.42550 


.72788 


1-37386 


•'75492 


1.32464 


.78269 


1.27764 


.81123 


1.23270 


57 


4 


.70194 


1.42462 


.72832 


1.37302 


.75538 


1.32384 


.78316 


1.27688 


.81171 


1. 23196 


56 


5 


.70238 


1.42374 


.72877 


1. 37218 


•75584 


1.32304 


.78363 


1.27611 


.81220 


1. 23123 


55 


6 


.70281 


1.42286 


.72921 


I.37I34 


•75629 


1.32224 


.78410 


1-27535 


.81268 


1.23050 


54 


7 


•70325 


1. 42198 


.72966 


1.37050 


•75675 


1. 32144 


•78457 


1.27458 


.81316 


1.22977 


53 


8 


.70368 


1.42110 


.73010 


1.36967 


.75721 


1.32064 


.78504 


1.27382 


•81364 


1 .22904 


52 


9 


.70412 


1.42022 


•73055 


1.36883 


.75767 


1. 31984 


.78551 


1 .27306 


.81413 


1. 22831 


5i 


IO 


.70455 


1. 41934 


.73100 


1 . 36800 


.75812 


1. 31904 


.78598 


1 .27230 


.81461 


1.22758 


50 


ii 


.70499 


1. 41847 


•73144 


1. 36716 


.75858 


1. 31825 


.78645 


1. 27153 


.81510 


1 .22685 


49 


12 


.70542 


I -4 I 759 


■73189 


1.36633 


•75904 


i-3i745 


.78692 


1.27077 


.81558 


1. 22612 


48 


x 3 


.70586 


1. 41672 


•73234 


1-36549 


.75950 


1. 31666 


.78739 


1 .27001 


.81606 


1.22539 


47 


14 


.70629 


1. 41584 


•73278 


1.36466 


.75996 


1. 31586 


.78786 


1.26925 


.81655 


1 .22467 


46 


15 


.70673 


1. 41497 


•73323 


1.36383 


.76042 


i-3 x 5C7 


.78834 


1.26849 


.81703 


1.22394 


45 


16 


.70717 


1. 41409 


•73368 


1 . 36300 


.76088 


1. 31427 


.78881 


1.26774 


.81752 


1. 22321 


44 


17 


.70760 


1. 41322 


•734!3 


1. 36217 


•76134 


1.31348 


.78928 


1.26698 


.81800 


1.22249 


43 


18 


. 70804 


!• 41235 


•73457 


1-36134 


.76180 


1. 31269 


.78975 


1.26622 


.81849 


1 .22176 


42 


i9 


.70848 


1.41148 


.73502 


1-36051 


. 76226 


1.31190 


.79022 


1.26546 


.81898 


1 .22104 


41 


20 


.70891 


1.41061 


•73547 


1.35968 


.76272 


1.31110 


.79070 


1. 26471 


.81946 


1. 2203 1 


40 


21 


.70935 


1.40974 


•73592 


1.35885 


.76318 


1.31031 


.79117 


1.26395 


.81995 


1. 21959 


39 


22 


.70979 


1.40887 


•73637 


1.35802 


.76364 


1.30952 


.79164 


1. 26319 


.82044 


1. 21886 


38 


23 


.71023 


1 . 40800 


.73681 


1. 35719 


.76410 


1.30873 


079212 


1 .26244 


.82092 


1.21814 


37 


24 


.71066 


1. 40714 


.73726 


I-35637 


.76456 


I-30795 


.79259 


1 .26169 


.82141 


1. 21742 


36 


25 


.71110 


1.40627 


•73771 


1-35554 


.76502 


1. 30716 


.79306 


1 .26093 


.82190 


1. 21670 


35 


26 


■7"54 


1 .40540 


.73816 


1.35472 


.76548 


1.30637 


•79354 


1. 26018 


.82238 


1. 21598 


34 


27 


.71198 


1.40454 


.73861 


1.35389 


.76594 


1-30558 


.79401 


1-25943 


.82287 


1. 21526 


33 


28 


.71242 


1.40367 


.73906 


I-35307 


.76640 


1 . 30480 


•79449 


1.25867 


.82336 


1. 21454 


32 


29 


.71285 


1. 40281 


.73951 


1.35224 


.76686 


1. 3040 1 


.79496 


1.25792 


.82385 


1. 21382 


3i 


3° 


.71329 


1 .40195 


.73996 


1. 35H2 


•76733 


1.30323 


•79544 


1. 25717 


•82434 


1.21310 


30 


3i 


•7 J 373 


1. 40109 


.74041 


1 .35060 


.76779 


1.30244 


.79591 


1.25642 


.82483 


1. 21238 


29 


32 


.71417 


1 .40022 


.74086 


1.34978 


.76825 


1. 30166 


.79639 


1-25567 


.82531 


1. 21166 


28 


33 


.71461 


!• 39936 


•74i3i 


1.34896 


.76871 


1.30087 


.796% 


1.25492 


.82580 


1. 21094 


27 


34 


•71505 


I.3985O 


.74176 


1. 34814 


.76918 


1 ,,30009 


•79734 


1-25417 


.82629 


1. 21023 


26 


35 


•71549 


I.39764 


.74221 


1-34732 


.76964 


1. 29931 


.7978i 


1-25343 


.82678 


1.20951 


25 


36 


•71593 


I.39679 


.74267 


1.34650 


.77010 


1.29853 


.79829 


1.25268 


.82727 


1 .20879 


24 


37 


•71637 


1 • 39593 


•74312 


1.34568 


•77057 


1.29775 


.79877 


1. 25193 


.82776 


1.20808 


23 


38 


.71681 


1 -395°7 


•74357 


1.34487 


•77 io 3 


1.29696 


.79924 


1.25118 


.82825 


1 .20736 


22 


39 


•71725 


1. 39421 


.74402 


1 . 34405 


• 77*49 


1. 29618 


.79972 


1.25044 


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1 . 20665 


21 


40 


.71769 


*• 39336 


•74447 


I-34323 


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1. 2954 1 


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1.24969 


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1.20593 


20 


4i 


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1.39250 


.74492 


1.34242 


.77242 


1.29463 


. 80067 


1.24895 


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1.20522 


19 


42 


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i.39 l6 5 


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1. 34160 


.77289 


1.29385 


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1.24820 


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1. 2045 1 


18 


43 


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1.39079 


•74583 


1.34079 


•77335 


1.29307 


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1.24746 


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1 .20379 


17 


44 


.71946 


1.38994 


.74628 


1.33998 


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1 .29229 


.80211 


1.24672 


.83120 


1.20308 


16 


45 


.71990 


1.38909 


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1. 33916 


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1. 29152 


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1.24597 


.83169 


1.20237 


15 


46 


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1.38824 


•74719 


I-33835 


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1.29074 


.80306 


1-24523 


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1 .20166 


14 


47 


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1.38738 


•74764 


1-33754 


•77521 


1.28997 


.80354 


1.24449 


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1 . 20095 


13 


48 


.72122 


1.38653 


.74810 


1-33673 


•77568 


1. 28919 


. 80402 


1-24375 


.83317 


1 .20024 


12 


49 


.72167 


1.38568 


.74855 


1-33592 


•77615 


1.28842 


.80450 


1 .24301 


.83366 


i-*9953 


11 


50 


.72211 


1.38484 


.74900 


1-335" 


.77661 


1.28764 


.80498 


1.24227 


•83415 


1. 19882 


IO 


5i 


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1-38399 


.74946 


I-33430 


.77708 


1.28687 


.80546 


1. 24153 


.83465 


1.19811 


9 


52 


.72299 


1. 38314 


.74991 


1-33349 


•77754 


1. 28610 


.80594 


1.24079 


.83514 


1. 1 9740 


8 


53 


•72344 


1.38229 


•75037 


1.33268 


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1-28533 


.80642 


1 . 24005 


.83564 


1. 19669 


7 


54 


.72388 


1-38145 


.75082 


1. 33187 


.77848 


1.28456 


. 8o6go 


1-23931 


.83613 


1 . 19599 


6 


55 


•72432 


1 . 38060 


.75128 


1. 33107 


•77895 


1.28379 


.80738 


1.23858 


.83662 


1. 19528 


5 


56 


.72477 


1.37976 


•75*73 


1.33026 


•77941 


1.28302 


.80786 


1.23784 


.83712 


1. 19457 


4 


57 


.72521 


1. 37891 


•75219 


1.32946 


.77988 


1.28225 


.80834 


1. 23710 


.83761 


1. 19387 


3 


58 


•72565 


1.37807 


.75264 


1.32865 


•78035 


1. 28148 


.80882 


1 .23637 


.83811 


1.19316 


2 


59 


.72610 


1.37722 


•753io 


1-32785 


.78082 


1. 28071 


.S0Q30 


1.23563 


.83S60 


1.19246 


X 


60 


.72654 


1-37638 


•75355 


1.32704 


.78129 


1.27994 


.80978 


1.23490 


.83910 


1-19175 





1 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


1 


5- 


*° 


5 


3° 


5 


2° 


5 


1° 


5 


D° 



G. G. IV 



40 



Natural Tangents and Cotangents. 



/ 


4 


D° 


■ 

4 


1° 


4 


2 S 


43° 


44° 


"I 

1 
60 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


o 


.83910 


1.19175 


.86929 


1-15037 


.90040 


1.11061 


.93252 


1.07237 


.96569 


1 -03553 


I 


.83960 


1.19105 


.86980 


1. 14969 


.90093 


1 . 10996 


.93306 


1. 07174 


.96625 


1.03493 


59 


2 


.84009 


1. 19035 


.87031 


1. 14902 


.90146 


1. 1093 1 


.93360 


1.07112 


.96681 


1 -°3433 


58 


3 


. 84059 


1. 18964 


.87082 


1. 14834 


.90199 


1. 10867 


•93415 


1.07049 


.96738 


1.03372 


57 


4 


.84108 


1. 18894 


.87133 


1. 14767 


.90251 


1 . 10802 


•93469 


1.06987 


•96794 


1. 03312 


56 


5 


.84158 


1.18824 


.87184 


1.14699 


.90304 


1. 10737 


•93524 


1.06925 


.96850 


1.03252 


55 


6 


.84208 


1.18754 


.87236 


1. 14632 


.90357 


1. 10672 


.93578 


1.06862 


.96907 


1-03192 


54 


7 


.84258 


1. 18684 


.87287 


1-14565 


.90410 


1 . 10607 


•93633 


1.06800 


.96963 


1. 03132 


53 


8 


.84307 


1.18614 


.87338 


1 . 14498 


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1. 10543 


.93688 


1.06738 


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1 .03072 


52 


9 


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1. 18544 


•87389 


1. 14430 


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1 . 10478 


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1.06676 


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1. 03012 


5i 


IO 


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1. 18474 


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1-14363 


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1.10414 


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1.06613 


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1.02952 


50 


ii 


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1 . 18404 


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1 . 14296 


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1 • 10349 


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1. 0655 1 


.97189 


1.02892 


49 


12 


.84507 


1. 18334 


.87543 


1. 14229 


.90674 


1. 10285 


.93906 


1.06489 


•97246 


1.02832 


48 


13 


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1. 18264 


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1.14162 


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1. 10220 


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1 .06427 


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1.02772 


47 


14 


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1.18194 


.87646 


1 . 14095 


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1.10156 


.94016 


1.06365 


•97359 


1. 02713 


46 


15 


.84656 


1. 18125 


.87698 


1. 14028 


.90834 


1. 1009 1 


.94071 


1.06303 


.97416 


1.02653 


45 


16 


.84706 


1. 18055 


•87749 


1.13961 


.90887 


1. 10027 


•94125 


1. 06241 


•97472 


1.02593 


44 


17 


.84756 


1. 17986 


.87801 


1. 13894 


.90940 


1.09963 


.94180 


1. 06179 


•97529 


1.02533 


43 


18 


.84806 


1.17916 


.87852 


1. 13828 


.90993 


1.09899 


•94235 


1.06117 


.97586 


1.02474 


42 


19 


.84856 


1. 17846 


.87904 


1.13761 


.91046 


1.09834 


.94290 


1 .06056 


•97643 


1.02414 


4i 


20 


. 84906 


1. 17777 


•87955 


1. 13694 


.91099 


1.09770 


•94345 


1.05994 


.97700 


1.02355 


40 


21 


.84956 


1. 17708 


.88007 


1. 13627 


• 9H53 


1 .09706 


.94400 


1.05932 


•97756 


1.02295 


39 


22 


.85006 


1. 17638 


.88059 


1.13561 


.91206 


1.09642 


•94455 


1 .05870 


•97813 


1.02236 


38 


23 


•85057 


1. 1 7569 


.88110 


1. 13494 


•91259 


1.09578 


.94510 


1.05809 


.97870 


1. 02176 


37 


24 


.85107 


1. 17500 


.88162 


1. 13428 


•91313 


1. 095 14 


•94565 


1-05747 


■97927 


1.02117 


36 


25 


•85157 


1. 17430 


.88214 


1.13361 


.01366 


1.09450 


.94620 


1.05685 


.97984 


1 .02057 


35 


26 


.85207 


1-17361 


.88265 


1. 13295 


.91419 


1.09386 


.94676 


1.05624 


.98041 


1. 01998 


34 


27 


.85257 


1. 17292 


.88317 


1. 13228 


•91473 


1.09322 


•94731 


1.05562 


.98098 


1. 01939 


33 


28 


•85308 


1. 17223 


.88369 


1.13162 


.91526 


1.09258 


.94786 


1. 05 501 


•98i55 


1. 01879 


32 


29 


.85358 


1-17154 


.88421 


1 . 1 3096 


.91580 


1. 09195 


.94841 


1.05439 


.98213 


1. 01820 


3i 


30 


.85408 


1. 1 7085 


.88473 


1 . 13029 


•91633 


1.09131 


.94896 


1.05378 


.98270 


1.01761 


30 


31 


.85458 


1.17016 


.88524 


1.12963 


.91687 


1.09067 


.94952 


1. 05317 


.98327 


1. 01702 


29 


32 


•85509 


1. 16947 


.88576 


1. 12897 


.91740 


1 . 09003 


.95007 


1-05255 


.98384 


1 .01642 


28 


33 


•85559 


1. 16878 


.88628 


1.12831 


.91794 


1 .08940 


.95062 


1. 05 194 


.98441 


1. 01583 


27 


34 


.85609 


1. 1 6809 


.88680 


1. 12765 


.91847 


1.08876 


.95118 


1-05133 


.98499 


1 .01524 


26 


35 


.85660 


1.16741 


.88732 


1.12699 


.91901 


1. 08813 


•95173 


1 .05072 


.98556 


1. 01465 


25 


36 


.85710 


1. 16672 


.88784 


1. 12633 


•91955 


1.08749 


•95229 


1 .05010 


.98613 


1. 01406 


24 


37 


.85761 


1. 1 6603 


.88836 


1. 12567 


.92008 


1.08686 


.95284 


1.04949 


.98671 


1. 01347 


23 


38 


.85811 


1-16535 


.88888 


1 . 12501 


.92062 


1 .08622 


•95340 


1.04888 


.98728 


1. 01288 


22 


39 


.85862 


1. 1 6466 


.88940 


1 .12435 


.92116 


1.08559 


•95395 


1.04827 


.98786 


1. 01229 


21 


40 


.85912 


1. 16398 


.88992 


1. 12369 


.92170 


1.08496 


•95451 


1.04766 


.98843 


1. 01170 


20 


4i 


.85963 


1. 16329 


.89045 


1. 12303 


.92224 


1.08432 


•955o6 


1.04705 


.98901 


1.01112 


19 


42 


.86014 


1. 56261 


.89097 


1. 12238 


.92277 


1.08369 


•95562 


1.04644 


.98958 


1 .01053 


18 


43 


.86064 


1.16192 


.89149 


1. 12172 


•92331 


1.08306 


.95618 


1.04583 


.99016 


1.00994 


17 


44 


.86115 


1.16124 


.89201 


1.12106 


.92385 


1.08243 


•95673 


1.04522 


.99073 


1.00935 


16 


45 


.86166 


1. 16056 


•89253 


1.12041 


•92439 


1. 08 1 79 


•95729 


1 .04461 


•99i3i 


1.00876 


15 


46 


.86216 


1. 15987 


.89306 


1.11975 


•92493 


1.08116 


•95785 


1. 04401 


.99189 


1. 00818 


14 


47 


.86267 


1.15919 


.89358 


1.11909 


•92547 


1.08053 


.95841 


1.04340 


.99247 


1.00759 


13 


48 


.86318 


1.15851 


.89410 


1. 1 1844 


.92601 


1.07990 


•95 8 97 


1.04279 


•993°4 


1 .00701 


12 


49 


.86368 


1-15783 


.89463 


1.11778 


•92655 


1.07927 


•95952 


1. 04218 


.99362 


1 .00642 


11 


50 


.86419 


i-i57iS 


.89515 


1.11713 


.92709 


1.07864 


.96008 


1. 04158 


.99420 


1.00583 


10 


5i 


.86470 


1.15647 


.89567 


1.1164S 


.92763 


1.07801 


. 96064 


1.04097 


.99478 


1.00525 


9 


52 


.86521 


1. 15579 


.89620 


1.11582 


1 .92817 


1.07738 


.96120 


1.04036 


.99536 


1.00467 


8 


53 


.86572 


1.15511 


.89672 


x. 1x5x7 


.92872 


1 .07676 


.96176 


1.03976 


•99594 


1 . 00408 


7 


54 


.86623 


1. 15443 


.89725 


1.11452 


.92926 


1 .07613 


.96232 


1. 03915 


.99652 


1.00350 


6 


55 


.86674 


I -i5375 


.89777 


1.11387 


.92980 


1-07550 


.96288 


1.03855 


.99710 


1 .00291 


5 


56 


.86725 


1.15308 


.89830 


1.11321 


•93034 


1.07487 


•96344 


1.03794 


.99768 


1.00233 


4 


57 


.86776 


1. 15240 


.89883 


1 .11256 


.93088 


1.07425 


.96400 


1.03734 


.99826 


1. 00175 


3 


58 


.86827 


1.15172 


•89935 


1.11191 


•93143 


1 .07362 


•96457 


1.03674 


.09884 


1.00116 


2 


59 


.86878 


1.15104 


.89988 


1.11126 


•93197 


1.07299 


•96513 


1. 03613 


•99942 


1.00058 


1 


60 


.86929 


1. 15037 


.90040 


1 .11061 


•93252 


1.07237 


.96569 


1-03553 


1 .00000 


1 .00000 





t 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


/ 


4 


9° 


4 


8° 


4 


7° 


4 


6° 


4 


5° 



TRAVERSE TABLES 

OR 

LATITUDES ^DEPARTURES OF COURSES 

CALCULATED TO 

THREE DECIMAL PLACES 

FOR 

EACH QUARTER DEGREE OF BEARING 



42 



LATITUDES AND DEPARTURES. 



to 

CO 

5" 


1 


2 


3 


4 


5 


*E-i 
<X5 

f=Q 


Lat. 


Dep. ] 


.at. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


0° 


I. OOO 


O.OOO 2 


OOO 


O.OOO 


3.000 


O.OOO 


4.000 


o.coo 


5.000 


90° 


°x 


I. OOO 


O.OO4 2 


OOO 


O.oog 


3.000 


0.013 


4.000 


0.017 


5.000 


89/ 


o/ 


I. OOO 


O.OO9 2 


OOO 


O.OI7 


3.000 


O.026 


4.000 


0.035 


5.000 


89/ 


°u 


I. OOO 


O.OI3 2 


OOO 


O.026 


3.OOO 


O.039 


4.000 


0.052 


5.000 


89/ 


1° 


I. OOO 


O.OI7 2 


OOO 


O.035 


3.000 


O.052 


3-999 


0.070 


4.999 


89° 


I* 


I. OOO 


0.022 2 


OOO 


O.044 


2.999 


O.065 


3-999 


0.0S7 


4-999 


88/ 


iK 


I. OOO 


O.O26 I 


999 


O.052 


2.999 


O.079 


3-999 


0.105 


4.998 


88/ 


i/ 


I. OOO 


0.031 I 


999 


O.061 


2.999 


O.O92 


3-998 


0.122 


4.998 


88/ 


2° 


0.999 


O.O35 I 


999 


9.070 


2.998 


O.I05 


3-99S 


0.140 


4-997 


88° 


2X 


0.999 


O.039 x 


998 


O.079 


2.998 


O.II8 


3-997 


0.157 


4.996 


87/ 


2/ 


0.999 


O.044 I 


998 


O.0S7 


2.997 


O.131 


3-99 6 


0.174 


4-995 


87/ 


23/ 


0.999 


O.048 I 


998 


O.096 


2.997 


O.144 


3-995 


0.192 


4.994 


87/ 


3° 


0.999 


O.052 I 


•997 


0.105 


2.996 


O.I57 


3-995 


0.209 


4-993 


8T° 


3/ 


0.99S 


O.057 I 


•997 


0.II3 


2-995 


O.170 


3-994 


0.227 


4.992 


86/ 


3K 


0.99S 


O.061 I 


.996 


O.I22 


2-994 


O.1S3 


3-993 


0.244 


4.991 


86}< 


3/ 


0.998 


O.065 I 


996 


0.I3I 


2.994 


O.I96 


3-991 


0.262 


4-989 


86/ 


4° 


0.998 


O.070 I 


995 


O.140 


2-993 


O.209 


3-990 


0.279 


4.9S8 


86° 


4/ 


0.997 


O.074 I 


•995 


O.14S 


2.992 


0.222 


3-9 s 9 


0.296 


4.986 


85/ 


4/ 


0.997 


O.078 I 


■994 


O.I57 


2.99I 


O.235 


3-9S8 


0.314 


4.9S5 


85/ 


4/ 
5° 


0.997 


O.0S3 I 


993 


O.166 


2.990 


0.248 


3-9S6 


0.331 


4-9S3 


85/ 


0.996 


O.087 * 


992 


O.I74 


2.9S9 


O.261 


3.9S5 


0.349 


4.9S1 


85° 


5/ 


0.996 


O.O92 1 


•992 


0.1 S3 


2.9S7 


O.275 


3.9S3 


0.366 


4-979 


84/ 


5/ 


0.995 


O.O96 I 


991 


0.192 


2.9S6 


O.2S8 


3-9S2 


0.3S3 


4-977 


84/ 


5/ 


0.995 


O. IOO I 


990 


0.200 


2.985 


O.30I 


3-98o 


0.401 


4-975 


84/ 


6° 


0.995 


O.I05 I 


9S9 


0.209 


2.9S4 


O.314 


3-978 


0.418 


4-973 


84° 


6/ 


0.994 


O.IO9 I 


.9S8 


0.218 


2.9S2 


O.327 


3.976 


0.435 


4.970 


S3/ 


6/ 


0.994 


O.II3 I 


.9S7 


0.226 


2.9S1 


O.340 


3-974 


0.453 


4.968 


83/ 


6/ 


0.993 


O.II8 I 


.986 


0.235 


2.979 


0-353 


3-972 


0.470 


4.965 


83/ 


r 


0.993 


O.I22 I 


935 


0.244 


2.978 


O.366 


3-97o 


0.4S7 


4-963 


83° 


7/ 


0.992 


O.I26 I 


.984 


0.252 


2.976 


0.379 


3-968 


0.505 


4.960 


82/ 


7/ 


0.991 


O.I3I I 


•983 


0.261 


2-974 


O.392 


3.966 


0.522 


4-957 


82/ 


7/ 


0.991 


O.I35 I 


.982 


0.270 


2-973 


O.405 


3-963 


0.539 


4-954 


82/ 


8° 


0.990 


O.I39 J 


.981 


0.278 


2.971 


O.418 


3.961 


0.557 


4-951 


82° 


8/ 


0.990 


O.I43 I 


•979 


0.2S7 


2.969 


O.430 


3-959 


0.574 


4.948 


81/ 


8/ 


0.9S9 


O.I48 I 


.978 


0.296 


2.967 


0.443 


3-956 


0.591 


4.945 


81/ 


5/ 


0.9SS 


O.I52 I 


•977 


0.304 


2.965 


O.456 


3-953 


0.608 


4.942 


81/ 


9 


0.988 


O.156 I 


•975 


0.313 


2.963 


O.469 


3-951 


0.626 


4-938 


81° 


9% 


0.987 


O.I6I I 


•974 


0.321 


2.961 


O.482 


3-94S 


0.643 


4-935 


80/ 


9 l A 


0.9S6 


O.165 I 


•973 


0.330 


2-959 


0.495 


3-945 


0.660 


4-931 


80/ 


9/ 
10° 


0.9S6 


O.169 I 


.971 


0.339 


2-957 


O.50S 


3-942 


0.677 


4.928 


80/ 
80° 


0.9S5 


O.174 I 


.970 


0.347 


2.954 


O.521 


3-939 


0.695 


4.924 


10/ 


0.984 


O.178 I 


.96S 


0.356 


2.952 


0-534 


3-936 


0.712 


4.920 


79/ 


10/ 


0.9S3 


O.IS2 I 


.967 


0.364 


2.950 


0.547 


3-933 


0.729 


4.916 


79/ 


10/ 


0.9S2 


O.1S7 I 


9^5 


0.373 


2-947 


O.560 


3-93Q 


0.746 


4.912 


79/ 


11° 


0.9S2 


O.I9I I 


9§3 


0.3S2 


2-945 


O.572 


3-927 


0.763 


4.908 


rr 


nX 


0.9SI 


O.I95 I 


962 


0.390 


2.942 


O.5S5 


3-9 2 3 


O.7S0 


4.904 


78/ 


nK 


0.9S0 


O.I99 x 


960 


0.399 


2.940 


O.598 


3.920 


0.797 


4.900 


78/ 


113/ 


0.979 


O.204 I 


958 


0.407 


2-937 


O.61 1 


3.916 


0.815 


4-895 


78/ 


12° 


0.973 


O.208 I 


956 


0.416 


2-934 


O.624 


3-9 T 3 


0.832 


4.891 


78° 


12* 


0.977 


0.2I2 I 


954 


0.424 


2.932 


O.637 


3-9°9 


0.849 


4.8S6 


773/ 


12/ 


0.976 


0.2I6 I 


953 


0.433 


2.929 


O.649 


3-9°5 


0.866 


4.881 


77/ 


123/ 


0.975 


0.22I I 


95i 


0.441 


2.926 


0.662 


3.901 


0.883 


4.877 


77* 


13 


0.974 


O.225 I 


949 


0.450 


2.Q23 


O.675 


3-S 9 7 


0.900 


4.872 


rr° 


CD 

1 

0O. 


Dep. 


Lat. I 


)ep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


pa 


1 


2 


3 


4 


5 


1 

F9 J 



LATITUDES AND DEPARTURES. 



43 



— 53- 

CD 

p? 

5" 

q©. 
0° 


5 


6 


7 


8 


9 


OS 

CQ 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


o.ooo 


6.000 


O.OOO 


7.000 


O.OOO 


8.000 


O.OOO 


9.000 


O.OOO 


90° 


o/ 


0.022 


6 


000 


O.026 


7.000 


O.031 


8.000 


0.035 


9 


000 


O.039 


89/ 


o^ 


O.O44 


6 


000 


O.052 


7.000 


O.061 


8.000 


0.070 


9 


000 


O.079 


s 9 y 2 


o/ 


O.065 


5 


999 


0.079 


6.999 


O.092 


7-999 


0.105 


8 


999 


O.II8 


89/ 


1° 


O.0S7 


5 


999 


0.105 


6.999 


O.I22 


7-999 


0.140 


8 


999 


O.I57 


89 


^K 


O.IO9 


5 


999 


O.13I 


6.998 


O.I53 


7.998 


O.I75 


S 


998 


O.I96 


883/ 


1% 


O.I3I 


5 


998 


O.I57 


6.998 


0.183 


7-997 


O.209 


8 


997 


O.236 


88/ 


*¥ 


O.I53 


5 


997 


0.183 


6.997 


O.214 


7.996 


O.244 


8 


996 


O.275 


88/ 


2° 


O.I74 


5 


996 


O.209 


6.996 


O.244 


7-995 


O.279 


8 


995 


O.314 


88° 


2 K 


O.I96 


5 


995 


O.236 


6.995 


0.275 


7-994 


O.314 


8 


993 


0.353 


87/ 


2J 4 


0.2I8 


5 


994 


O.262 


6-993 


0.305 


7.992 


0.349 


8 


991 


0-393 


87/ 


2/ 


O.24O 


5 


993 


O.288 


6.992 


0.336 


7.991 


O.384 


8 


990 


0.432 


87/ 


3° 


O.262 


5 


992 


O.314 


6.990 


O.366 


7.989 


0.419 


S 


988 


0.47I 


8V 


3* 


O.283 


5 


990 


O.340 


6.989 


0.397 


7.987 


0.454 


s 


986 


O.510 


863/ 


3/ 


0.305 


5 


989 


0.366 


6.987 


O.427 


7.985 


O.488 


8 


983 


0.549 


86/ 


3/ 


O.327 


5 


987 


O.392 


6.985 


O.458 


7.983 


O.523 


8 


981 


O.589 


86X 
86° 


4° 


0.349 


5 


985 


O.419 


6.983 


O.488 


7.981 


o.558 


8 


978 


O.628 


4* 


O.37I 


5 


984 


0.445 


6.981 


O.519 


7.978 


0-593 


8 


975 


O.667 


*o s H 


4K 


O.392 


5 


982 


O.471 


6.978 


0.549 


•975 


O.628 


8 


972 


O.706 


85K 


4/ 
5° 


O.414 


5 


979 


o.497 


6.976 


0.580 


7-973 


O.662 


8 


969 


0.745 


85/ 

85° 


O.436 


5 


977 


0.523 


6-973 


0.610 


7.970 


0.697 


8 


966 


O.784 


5/ 


0.453 


5 


975 


0.549 


6.971 


O.641 


7.966 


o.732 


8 


962 


O.824 


843/ 


5/ 


O.479 


5 


972 


0.575 


6.968 


O.671 


7.963 


O.767 


8 


959 


O.863 


84/ 


53/ 


O.50I 


5 


97o 


0.601 


6.965 


0.701 


7.960 


0.802 


8 


955 


O.902 


84/ 


6° 


0.523 


5 


967 


O.627 


6.962 


0.732 


7.956 


0.836 


S 


95i 


O.941 


84° 


6/ 


0-544 


5 


964 


O.653 


6.958 


O.762 


7-952 


0.871 


8 


947 


O.980 


83/ 


6/ 


O.566 


5 


961 


0.679 


6-955 


0.792 


7-949 


O.906 


8 


942 


I. Olg 


83/ 


6 ^ 


O.588 


5 


953 


O.705 


6.951 


O.823 


7-945 


0.940 


8 


938 


I.058 


83/ 


O.609 


5 


955 


O.731 


6.948 


O.853 


7.940 


0.975 


8 


933 


I.097 


83° 


7^r 


O.63I 


5 


952 


0.757 


6.944 


0.883 


7-936 


1. 010 


8 


928 


LI36 


32/ 


7/ 


O.653 


5 


949 


O.783 


6.940 


O.914 


7-932 


I.044 


8 


923 


I.I75 


82/ 


73/ 


O.674 


5 


945 


0.809 


6.936 


O.944 


7.927 


I.079 


8 


918 


1. 214 


82/ 


8° 


O.696 


5 


942 


0.835 


6.932 


0.974 


7.922 


1. 113 


S 


912 


1-253 


82° 


8/ 


O.717 


5 


938 


0.861 


6.928 


I.004 


7.917 


1. 148 


S 


907 


1. 291 


81/ 


8/ 


0.739 


5 


934 


O.887 


6.923 


I.035 


7.912 


1. 182 


s 


901 


I-330 


si/ 


8* 


O.761 


5 


93o 


O.913 


6.919 


I.065 


7.907 


1. 217 


s 


895 


1.369 


81/ 


9° 


O.782 


5 


926 


0.939 


6.914 


I.095 


7.902 


1. 251 


s 


8S9 


1.408 


81° 


9/ 


O.804 


5 


922 


O.964 


6.909 


I.125 


7.896 


I.286 


8 


883 


1.447 


80/ 


9/ 


O.825 


5 


918 


O.990 


6.904 


I-I55 


7.890 


I.320 


8 


877 


1.485 


So/ 


9 3 / 


O.847 


5 


913 


T.016 


6.899 


1. 185 


7.884 


1-355 


8 


870 


1.524 


So/ 
80° 


10° 


O.868 


5 


909 


I.042 


6.894 


1. 216 


7.878 


1.389 


8 


863 


1-563 


io/ 


O.89O 


5 


9°4 


I.068 


6.888 


1.246 


7.872 


1.424 


S 


S56 


i.6or 


79/ 


10/ 


O.9II 


5 


900 


I.093 


6.883 


1.276 


7.866 


1.458 


8 


S49 


1.640 


79/ 


103/ 


0.933 


5 


8q5 


1. 119 


6.877 


1.306 


7.860 


1.492 


8 


S42 


1.679 


79/ 


11° 


0.954 


5 


890 


I.I45 


6.871 


1-336 


7.853 


1.526 


8 


335 


1. 717 


79 


II/ 


0.975 


5 


885 


1. 171 


6.866 


1.366 


7.846 


1.561 


S 


S27 


1.756 


78/ 


"K 


0.997 


5 


.880 


1. 196 


6.859 


1.396 


7.839 


1-595 


8 


S19 


1.794 


7S/ 


113/ 


I. Ol8 


5 


874 


1.222 


6.853 


1.425 


7.S32 


1.629 


8 


Sn 


1-833 


7«# 


12° 


T.04O 


5 


869 


1.247 


6.847 


1-455 


7.825 


1.663 


8 


S03 


1.S71 


rs 


12/ 


1. 061 


5 


863 


1.273 


6.841 


1.4S5 


7.81S 


1.697 


S 


795 


1. 910. 


77/ 


12^ 


I.082 


5 


858 


I.299 


6.834 


1. 5i5 


7-Sro 


1.732 


s 


7S7 


1-948 


7 / . 2 


123/ 


1. 103 


5 


852 


1-324 


6.827 


1-545 


7-803 


1.766 


s 


773 


1.9S6 


I~ I+ 


13° 

OO. 


1. 125 


5 


846 


T-350 


6.821 


1-575 


7-795 


1.800 


s 


769 


2.025 


rr- 


Lat. 


I 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lac. 


Dep. 


Lat. 


5 


6 


7 


8 


9 



44 



LATITUDES AND DEPARTURES. 



CO 

CD 
P3 

5" 
13° 


1 


2 


3 


4 


5 


•ob 
QQ 

rr 


Lat. 


Dep. I 


.at. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


0.974 


O.225 T 


949 


O.450 


2.923 


O.675 


3-897 


O.9OO 


4.872 


13* 


0.973 


O.229 I 


947 


O.458 


2.920 


O.688 


3.894 


O.917 


4.867 


763/ 


1.3* 


O.972 


O.233 I 


945 


O.467 


2.917 


O.700 


3-889 


0-934 


4.862 


76* 


1334: 


O.971 


O.238 I 


943 


0.475 


2.914 


O.713 


3.885 


0.95I 


4.857 


76 % 


14 


O.970 


O.242 I 


941 


O.4S4 


2. 91 1 


O.726 


3-88i 


O.968 


4.851 


76° 


T4^ 


O.969 


O.246 I 


938 


O.492 


2.908 


0.738 


3-877 


O.985 


4.846 


753/ 


14* 


0.968 


O.250 I 


936 


O.50I 


2.904 


0.75I 


3-873 


I.002 


4.841 


75* 


143/ 

15° 


O.967 


O.255 I 


934 


O.509 


2.9OI 


O.764 


3.86S 


1.01S 


4.835 


75^ 
75° 


O.966 


O.259 I 


932 


O.518 


2.89S 


0.776 


3.S64 


1.035 


4.830 


15* 


O.965 


O.263 * 


930 


O.526 


2.894 


O.789 


3.859 


1.052 


4.824 


743/ 


15* 


O.964 


O.267 1 


927 


0.534 


2.89I 


O.802 


3-855 


1.069 


4.818 


74* 


153^ 


O.962 


O.271 I 


925 


0.543 


2.8S7 


O.814 


3.850 


1.086 


4.812 


74 % 


16° 


O.961 


O.276 I 


923 


0.55I 


2.884 


O.827 


3-845 


1-103 


4.806 


w 


16* 


O.960 


O.280 I 


920 


O.560 


2.8SO 


O.839 


3.840 


r.119 


4.800 


733/ 


16* 


0-959 


O.2S4 I 


918 


O.568 


2.876 


O.852 


3.835 


1. 136 


4-794 


73* 


163/ 


O.958 


O.288 I 


915 


O.576 


2.873 


O.865 


3.830 


I.I53 


4.788 


73X 


17° 


O.956 


O.292 I 


913 


0.585 


2.869 


O.877 


3-825 


1. 169 


4.782 


73° 


17* 


0-955 


O.297 I 


910 


0.593 


2.865 


O.890 


3.820 


1.186 


4-775 


72 3/ 


17* 


0.954 


O.301 I 


907 


O.601 


2.861 


O.902 


3-8i5 


1.203 


4.769 


72* 


173/ 


O.952 


O.305 I 


9°5 


O.610 


2.857 


O.915 


3.810 


1.220 


4.762 


72* 


18° 


O.951 


0. 309 I 


902 


O.618 


2.853 


O.927 


3.804 


1.236 


4-755 


72° 


18* 


O.950 


O.313 -I 


899 


O.626 


2.849 


0.939 


3-799 


1.253 


4.748 


713/ 


IB* 


O.948 


O.317 1 


897 


O.635 


2.845 


O.952 


3-793 


1.269 


4.742 


7i* 


183/ 


0.947 


O.321 I 


894 


O.643 


2.84I 


O.964 


3.788 


1.286 


4-735 


7i % 


19° 


O.946 


O.326 I 


891 


O.651 


2.837 


O.977 


3.782 


1.302 


4.728 


n° 


19* 


O.944 


O.330 I 


888 


O.659 


2.832 


O.989 


3-776 


I-3I9 


4.720 


703/ 


19* 


0-943 


0.334 I 


885 


O.668 


2.828 


I. OOI 


3-771 


1-335 


4.713 


70* 


193/ 
20 3 


O.941 


O.338 I 


882 


O.676 


2.824 


1. 014 


3.765 


1-352 


4.706 


70X 
70° 


O.940 


O.342 I 


879 


O.6S4 


2.819 


I.026 


3-759 


1.368 


4.698 


20X 


O.938 


O.346 I 


876 


O.692 


2.815 


I.038 


3-753 


1.384 


4.691 


693/ 


20* 


0.937 


0.350 I 


873 


O.700 


2.8IO 


1. 051 


3-747 


1. 401 


4-683 


69* 


20 3/ 


0-935 


0-354 I 


870 


O.709 


2.805 


I.063 


3-741 


I-4I7 


4.676 


69* 


21° 


0-934 


O.358 I 


867 


O.717 


2.80I 


i.o75 


3-734 


1-433 


4.668 


69° 


21* 


O.932 


O.362 I 


864 


O.725 


2.796 


1.087 


3.728 


1.450 


4.660 


683/ 


21* 


O.930 


O.367 I 


861 


0.733 


2.79I 


1. 100 


3-722 


1.466 


4-652 


68* 


213/ 


O.929 


O.371 I 


858 


O.741 


2.786 


1.112 


3-715 


1.482 


4.644 


b^4 


22 J 


O.927 


0-375 1 


■854 


O.749 


2.782 


1. 124 


3-709 


1.498 


4.636 


68° 


22X 


O.926 


o.379 1 


.851 


0.757 


2.777 


1. 136 


3.702 


I-5I5 


4.628 


673/ 


22* 


O.924 


0.383 1 


848 


0.765 


2.772 


1.14S 


3.696 


I-53I 


4.619 


67* 


22 3/ 


O.922 


0.387 1 


.844 


0.773 


2.767 


1. 160 


3-689 


1-547 


4.611 


67^ 


23 


O.921 


0.391 1 


.841 


O.781 


2.762 


1. 172 


3.682 


1-563 


4.603 


67° 


23^ 


O.919 


o.395 1 


.838 


O.789 


2.756 


1. 184 


3.675 


1-579 


4-594 


663/ 


23* 


O.917 


o.399 1 


.834 


0.797 


2.751 


1. 196 


3.66S 


1-595 


4.585 


66* 


23 3/ 


O.915 


0.403 1 


.831 


O.805 


2.746 


1.208 


3.661 


1. 611 


4-577 


66^ 


24 ' 


O.914 


0.407 1 


.827 


O.813 


2.74I 


1.220 


3-654 


1.627 


4-568 


66° 


24^ 


O.912 


0.411 1 


824 


O.821 


2.735 


1.232 


3-647 


1.643 


4-559 


653/ 


24* 


O.910 


0.415 1 


820 


O.829 


2.730 


1.244 


3.640 


1.659 


4-550 


65* 


243/ 
25 ' 


O.908 


0.419 1 


816 


O.837 


2.724 


1.256 


3-633 


1.675 


4-541 


65X 
65° 


O.906 


0.423 1 


813 


O.845 


2.719 


1.268 


3.625 


1.690 


4-532 


2$X 


O.904 


0.427 1 


809 


O.853 


2.713 


1.280 


3.618 


1.706 


4-522 


643/ 


25/2 


O.903 


0.431 1 


805 


O.861 


2.708 


1.292 


3.610 


1.722 


4-513 


64* 


253/ 


O.901 


0-434 1 


801 


O.869 


2.702 


1-303 


3.603 


1.738- 


4-503 


64. 1 / 


26° 

5. 

5 
00. 


O.899 


0.438 1 


798 


O.877 


2.696 


i-3i5 


3-595 


1-753 


4.494 


64° 


Dep. 


Lat. I 


)ep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


■ob 

a 

c3 

133 


1 


2 


3 


4 


5 



LATITUDES AND DEPARTURES. 



45 



CD 


5 


6 


7 


8 


9 




3* 

qo. 

13° 












erf 

CO 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


1. 125 


5-846 


I-35Q 


6. 82 1 


1-575 


7-795 


I.800 


8.769 


2.025 


XX° 


iv4 


I.146 


5 


840 


1-375 


6.814 


I.604 


7.787 


I.834 


8.760 


2.063 


763/ 


13^ 


I.167 


5 


834 


1. 401 


6.807 


I.634 


7-779 


1.868 


8-751 


2. IOI 


76^ 


13 3/ 


1. 188 


3 


828 


1.426 


6-799 


T.664 


7-771 


1.902 


8.742 


2.139 


76/ 


14 


1. 2IO 


5 


822 


1.452 


6.792 


I.693 


7.762 


1-935 


8-733 


2-177 


76 


14^ 


1. 231 


5 


815 


1-477 


6.7S5 


I.723 


7-754 


1.969 


8.723 


2.215 


753/ 


14* 


I.252 


5 


809 


1.502 


6-777 


1-753 


7-745 


2.003 


8.713 


2.253 


75K 


14^ 
15 


I.273 


5 


802 


1.528 


6.769 


I.782 


7-736 


2.037 


8.703 


2.291 


75/ 
7S° 


1-294 


5 


796 


1-553 


6.761 


1. 812 


7.727 


2.071 


8.693 


2.329 


15* 


I-3I5 


5 


789 


1.578 


6-754 


1. 841 


7.718 


2.104 


8.683 


2.367 


743/ 


15* 


1-336 


5 


782 


1.603 


6-745 


1. 871 


7.709 


2.138 


8.673 


2.405 


74 Vz 


i5# 


1-357 


5 


775 


1.629 


6-737 


I.900 


7.700 


2.172 


8.662 


2-443 


74/ 


16° 


1-378 


5 


768 


1.654 


6.729 


I.929 


7.690 


2.205 


8.651 


2.481 


?4 D 


16* 


1-399 


5 


760 


1.679 


6.720 


1-959 


7.680 


2.239 


8.640 


2.518 


733/ 


16* 


1.420 


5 


753 


1.704 


6.712 


1.988 


7.671 


2.272 


8.629 


2.556 


73K 


l6tf 


1. 441 


5 


745 


1.729 


6.703 


2.017 


7.661 


2.306 


8.618 


2-594 


73/ 


ir 


1.462 


5 


738 


1-754 


6.694 


2.047 


7.650 


2-339 


8.607 


2.631 


73° 


17* 


1.483 


5 


730 


1-779 


6.6S5 


2.076 


7.640 


2.372 


8-595 


2.669 


72/ 


17* 


1.504 


5 


722 


1.804 


6.676 


2.105 


7.630 


2.406 


8.583 


2.706 


72^ 


i7 3 / 


1.524 


5 


714 


1.829 


6.667 


2.134 


7.619 


2-439 


8.572 


2.744 


72^ 


18 


1-545 


5 


706 


1.854 


6.657 


2.163 


7.608 


2.472 


8.560 


2.781 


72° 


i8tf 


1.566 


5 


698 


1.879 


6.648 


2.192 


7-598 


2.505 


8-547 


2.818 


71 u 


18* 


1.587 


5 


690 


1.904 


6.638 


2.221 


7-5*7 


2.538 


8-535 


2.856 


71 >£ 


I»tf 


1.607 


5 


682 


1.929 


6.629 


2.250 


7-575 


2-572 


8.522 


2.893 


71X 


19° 


1.628 


5 


673 


1-953 


6.619 


2.279 


7.564 


2.605 


8.5IO 


2.930 


XV 


1 9 * 


1.648 


5 


665 


1.978 


6.609 


2.308 


7-553 


2.638 


8-497 


2.967 


70 / 


19^ 


1.669 


5 


656 


2.003 


6.598 


2.337 


7-541 


2.670 


8.484 


3.OO4 


70/ 


19% 
20 


1.690 


5 


647 


2.028 


6.588 


2.365 


7-529 


2.703 


8.471 


3.041 


70 / 

70° 


1. 710 


5 


638 


2.052 


6.578 


2-394 


7.518 


2.736 


8-457 


3.078 


20^ 


I-73I 


5 


629 


2.077 


6.567 


2.423 


7.506 


2.769 


8.444 


3-H5 


69% 


20^ 


1. 75i 


5 


620 


2.101 


6-557 


2.451 


7-493 


2.802 


8.430 


3-152 


69 Vz 


203/ 


T.771 


5 


611 


2.126 


6.546 


2.480 


7.481 


2.834 


8.416 


3.189 


69/ 


21° 


1.792 


5 


601 


2.150 


6-535 


2.509 


7.469 


2.867 


8.402 


3.225 


69° 


21X 


1. 812 


5 


592 


2-175 


6.524 


2-537 


7-456 


2.900 


8.388 


3.262 


63 3/ 


21K 


1-833 


5 


582 


2.199 


6.513 


2.566 


7-443 


2.932 


8-374 


3-299 


63 y 2 


213^ 


1-853 


5 


573 


2.223 


6.502 


2-594 


7-43Q 


2.964 


8-359 


3-335 


68/ 


22° 


1.873 


5 


563 


2.248 


6.490 


2.622 


7-417 


2-997 


8-345 


3-371 


68° 


22* 


1.893 


5 


553 


2.272 


6-479 


2.651 


7.404 


3.029 


8.330 


3.408 


673/ 


iiy z 


I-9I3 


5 


543 


2.296 


6.467 


2.679 


7-39 1 


3.061 


8.315 


3-444 


67/ 2 


223^ 


1-934 


5 


533 


2.320 


6-455 


2.707 


7-378 


3.094 


8.300 


3.480 


b V4 


23 


1-954 


5 


523 


2-344 


6.444 


2-735 


7-364 


3.126 


8.285 


3-517 


67° 


23X 


1.974 


5 


513 


2.368 


6.432 


2.763 


7-35Q 


3-158 


8.269 


3-553 


663/ 


23 % 


1.994 


5 


502 


2.392 


6.419 


2.791 


7-336 


3.190 


8.254 


3.589 


66 y, 


23 3 / 


2.014 


5 


492 


2.416 


6.407 


2.819 


7.322 


3.222 


8.23S 


3.625 


bb% 


24 


2.034 


5 


481 


2.440 


6-395 


2.847 


7-308- 


3-254 


8.222 


3.661 


66° 


24* 


2.054 


5 


47i 


2.464 


6.382 


2.875 


7.294 


3.2S6 


8.206 


3.696 


65/ 


24K 


2.073 


5 


460 


2.488 


6.370 


2.903 


7.280 


3.318 


8.I9O 


3-732 


65K 


24M 
25 


2.093 


5 


449 


2.512 


6-357 


2.931 


7.265 


3-349 


3.173 


3.76S 


65 


2. 113 


5 


438 


2.536 


6-344 


2.95S 


7.250 


3-3SI 


S.I57 


3-804 


25X 


2.133 


5 


427 


2-559 


6.331 


2.9S6 


7.236 


3-4T3 


S.I40 


3-S39 


04 u 


25K 


2.153 


5 


416 


2.583 


6.318 


3.014 


7.221 


3-4-14 


S.I23 


3-375 


04 >, 


253/ 


2.172 


5 


404 


2.607 


6.305 


3.041 


7.206 


3-476 


8. 106 


3.910 


04 V 


26 

DO 

CD 
P= 

5" 
00, 


2.192 


5 


393 


2.630 


6.292 


3.069 


7.190 


3-507 


S.089 


3-945 


64 

a 

ga 
_PQ 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


5 


6 


7 


8 


9 



46 



LATITUDES AND DEPARTURES. 



5" 

O©. 

26° 


1 


2 


3 


4 


5 


"OO 

PQ 

64° 


Lat. 


Dep. I 


.at. 


Dep. 


Lat. I 


)ep. 


Lat. 


Dep. 


Lat. 


O.899 


0.438 I 


798 


O.877 


2.696 I 


315 


3-595 


1-753 


4.494 


26^ 


O.897 


O.442 I 


794 


O.885 


2.69I I 


327 


3-5S7 


1.769 


4.484 


63^ 


26^ 


O.895 


O.446 I 


790 


O.892 


2.685 I 


339 


3-58o 


1-785 


4-475 


63 X 


263/ 


O.893 


O.450 I 


7S6 


O.9OO 


2.679 x 


35o 


3-572 


1.800 


4-465 


63 % 


2r 


O.89I 


0-454 I 


782 


O.908 


2.673 I 


362 


3-564 


1.816 


4-455 


63° 


27^ 


O.889 


O.458 I 


778 


O.916 


2.667 I 


374 


3-556 


1. 831 


4-445 


62 3/ 


27K 


0.887 


O.462 I 


774 


O.923 


2.66l I 


385 


3-548 


1.847 


4-435 


62 y 2 


273/ 


O.885 


O.466 I 


770 


O.931 


2.655 I 


397 


3-54o 


1.862 


4-42 5 


62X 


28> 


O.8S3 


O.469 I 


766 


0-939 


2.649 I 


408 


3-532 


1.878 


4-415 


62° 


28^ 


O.881 


0-473 I 


762 


0.947 


2.643 I 


420 


3-524 


1.893 


4.404 


613/ 


2sy 2 


O.879 


O.477 I 


758 


0.954 


2.636 I 


43i 


3-515 


1.909 


4-394 


6i# 


283/ 


O.877 


O.481 I 


753 


O.962 


2.630 I 


443 


3-507 


1.924 


4-384 


61X 


29° 


O.875 


O.485 I 


749 


O.970 


2.624 I 


454 


3-498 


!-939 


4-373 


61° 


29X 


O.872 


O.489 I 


745 


0-977 


2.617 I 


466 


3-490 


1-954 


4.362 


603/ 


29K 


O.870 


O.492 I 


74i 


O.985 


2. 6ll I 


477 


3.481 


1.970 


4-352 


60 y 2 


293/ 

30° 


O.868 


O.496 I 


736 


O.992 


2.605 I 


489 


3-473 


1.985 


4-341 
4-330 


60X 

60° 


O.866 


0. 500 I 


732 


I. OOO 


2.598 I 


500 


3-464 


2.000 


3o^ 


O.864 


0. 504 I 


728 


I.008 


2.592 I 


5ii 


3-455 


2.015 


4-319 


59* 


30K 


O.862 


O.508 I 


723 


I. OI5 


2.585 I 


523 


3-447 


2.030 


4.308 


59K 


30 3/ 


O.859 


0. 511 I 


719 


I.023 


2:578 I 


534 


3-438 


2.045 


4.297 


59X 


31° 


O.857 


0.5I5 I 


7i4 


I.030 


2.572 I 


545 


3-429 


2.060 


4.286 


59° 


3iX 


O.855 


O.519 I 


710 


I.038 


2.565 I 


556 


3.420 


2.075 


4-275 


58^ 


■31 K 


O.853 


O.522 I 


705 


I.045 


2.558 I 


567 


3-4H 


2.090 


4.263 


58^ 


31 3 / 


O.850 


O.526 I 


701 


I.052 


2-551 1 


579 


3.401 


2.105 


4-252 


58/ 


32° 


O.848 


O.530 I 


696 


I.060 


2.544 1 


59° 


3-392 


2.120 


4.240 


58° 


32X 


O.846 


0.534 I 


691 


I.067 


2-537 I 


601 


3-383 


2.134 


4.229 


571/ 


32^ 


O.843 


0-537 I 


6S7 


I.075 


2.530 * 


612 


3-374 


2.149 


4.217 


57K 


32/ 


O.841 


0.541 I 


682 


I.082 


2.523 I 


623 


3-364 


2.164 


4-205 


SlU 


33 


O.839 


0-545 I 


677 


I.089 


2.516 I 


634 


3-355 


2.179 


4-193 


sr 


33# 


O.836 


O.548 I 


673 


I.097 


2.509 I 


645 


3-345 


2.193 


4.181 


56X 


33K 


O.834 


0.552 I 


668 


1. 104 


2.502 I 


656 


3-336 


2.208 


4.169 


56^ 


33 U 


O.831 


0.556 I 


663 


I. Ill 


2.494 I 


667 


3-326 


2.222 


4-157 


56^ 


34 


O.829 


0.559 I 


658 


I.II8 


2.487 I 


.678 


3-3i6 


2.237 


4-145 


56° 


34X 


O.S27 


O.563 I 


653 


1. 126 


2.480 I 


688 


3- 3°6 


2.251 


4-133 


553/ 


34K 


O.824 


O.566 I 


.648 


I.I33 


2.472 I 


.699 


3-297 


2.266 


4.121 


ssy 2 


34* 
35 


O.822 


0.570 I 


643 


1. 140 


2.465 I 


710 


3.287 


2.280 


4.108 


55X 
55° 


O.819 


0.574 I 


.638 


1. 147 


2-457 I 


.721 


3-277 


2.294 


4.096 


35^ 


O.817 


0.577 I 


•633 


I.I54 


2.450 I 


•731 


3.267 


2.309 


4.083 


54X 


35K 


O.814 


O.581 I 


.628 


I.l6l 


2.442 I 


.742 


3-257 


2.323 


4.071 


54^ 


353/ 


O.812 


O.584 I 


•623 


I.I68 


2-435 I 


•753 


3.246 


2-337 


4-058 


54^ 


36 


O.809 


O.588 I 


.618 


I.176 


2.427 I 


•763 


3-236 


2-351 


4-Q45 


54° 


36X 


O.806 


0.591 I 


•613 


I.1S3 


2.419 I 


774 


3.226 


2.365 


4.032 


53M 


36^ 


O.804 


0.595 I 


.608 


1. 190 


2.412 I 


.784 


3215 


2-379 


4.019 


53^ 


3 £K 


O.801 


O.598 I 


.603 


.1.197 


2.404 I 


•795 


3-205 


2-393 


4.006 


53X 


37 


O.799 


0. 602 I 


•597 


I.204 


2.396 I 


S05 


3-195 


2.407 


3-993 


53° 


37^ 


O.796 


0. 605 I 


•592 


I. 211 


2.388 I 


816 


3.184 


2.421 


3.980 


523/ 


37K 


0.793 


0. 609 I 


.5S7 


I. 218 


2.3S0 I 


826 


3-173 


2-435 


3-967 


52^ 


37* 


O.791 


O.612 I 


.581 


1.224 


2.372 I 


837 


3-163 


2.449 


3-953 


52X 


38° 


O.788 


O.616 I 


576 


I. 231 


2.364 I 


847 


3-152 


2.463 


3-94o 


52° 


38>( 


O.785 


0.619^ I 


•57i 


I.238 


2.356 I 


857 


3-I4I 


2.476 


3-927 


513/ 


38M 


O.783 


O.623 I 


565 


I.245 


2.34S I 


868 


3-130 


2.490 


3-9*3 


5i^ 


383/ 


O.780 


O.626 I 


560 


1.252 


2.340 I 


878 


3.120 J 2.504 


3-899 


5iX 


39 

do 

CD 

5" 
OO. 


0.777 


O.629 I 


554 


I.259 


2.331 I 


8S8 


3.109 


2.517 


3.886 


51° 

(=3 

"5 

<x> 


Dep. 


Lat. I 


)ep. 


Lat. 


Dep. ] 


Lat. 


Dep. 


Lat. 


Dep. 


1 


2 


3 


4 


5 



LATITUDES AND DEPARTURES. 



4? 



5" 

qo. 
26 n 


5 


6 


7 


8 


9 


CO 

64° 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


2.192 


5-393 


2.630 


6.292 


3.069 


7.190 


3-507 


8.089 


3-945 


26/ 


2.21 1 


5-38i 


2.654 


6.278 


3.096 


7-175 


3.538 


8.o7j2 


3.981 


633/ 


26^ 


2.231 


5-37Q 


2.677 


6.265 


3-123 


7.160 


3-570 


8.054 


4.016 


63 y 2 


26/ 


2.250 


5-358 


2.701 


6.251 


3-I5I 


7.144 


3.601 


8.037 


4.051 


63^ 


27° 


2.270 


5-346 


2.724 


6.237 


3-178 


7.128 


3-632 


8.0T9 


4.086 


63° 


27^ 


2.289 


5-334 


2.747 


6.223 


3-205 


7. 1 12 


3.663 


8.001 


4.121 


62 3/ 


27^ 


2.309 


5-322 


2.770 


6.209 


3.232 


7.096 


3.694 


7-983 


4.T56 


62^ 


273/ 


2.328 


5-3io 


2.794 


6.195 


3-259 


7.080 


3-725 


7.965 


4.190 


62/ 


28° 


2-347 


5-298 


2.817 


6. 181 


3.286 


7.064 


3-756 


7-947 


4.225 


62 


28/ 


2.367 


5-285 


2.840 


6.166 


3-3*3 


7.047 


3-787 


7.928 


4.260 


61/ 


28K 


2.386 


5-273 


2.863 


6.152 


3-340 


7.031 


3-817 


7.909 


4.294 


61K 


283/ 


2.405 


5.260 


2.886 


6.137 


3-367 


7.014 


3-848 


7.891 


4.329 


61X 


29° 


2.424 


5-248 


2.909 


6.122 


3-394 


6-997 


3.878 


7.872 


4-363 


61° 


29^ 


2-443 


5-235 


2.932 


6.107 


3.420 


6.980 


3-9°9 


7.852 


4-398 


603/ 


29K 


2.462 


5.222 


2-955 


6.093 


3-447 


6.963 


3-939 


7-833 


4-432 


60^ 


29^ 
30° 


2.481 


5-209 


2-977 


6.077 


3-474 


6.946 


3-97Q 


7.814 


4.466 


60X 
60° 


2.500 


5.196 


3.000 


6.062 


3-5oo 


6.928 


4.000 


7-794 


4-5oo 


30^ 


2.519 


5-i83 


3-023 


6.047 


3-526 


6.911 


4.030 


7-775 


4-534 


59^ 


30^ 


2.538 


5-i7o 


3-Q45 


6.031 


3-553 


6.893 


4.060 


7-755 


4-568 


59^ 


30 3/ 


2.556 


5-I56 


3.068 


6.016 


3-579 


6.875 


4.090 


7-735 


4.602 


S9 1 / 


31° 


2-575 


5-143 


3.090 


6.000 


3.605 


6.857 


4.120 


7-715 


4-635 


59° 


3i^ 


2-594 


5.129 


3- "3 


5-984 


3.631 


6.839 


4.150 


7.694 


4.669 


583/ 


3* J A 


2.612 


5 .ii6 


3-135 


5.968 


3-657 


6.821 


4.180 


7.674 


4.702 


58^ 


3 i# 


2.63I 


5.102 


3-157 


5-952 


3.683 


6.803 


4.210 


7-653 


4-736 


58/ 


2.650 


5.088 


3.180 


5.936 


3-709 


6.784 


4-239 


7.632 


4.769 


58° 


32^ 


2.668 


5-o74 


3.202 


5.920 


3-735 


6.766 


4.269 


7.612 


4.802 


571/ 


32^ 


2.686 


5.060 


3.224 


5.904 


3.761 


6.747 


4.298 


7-59 1 


4.836 


57^ 


32 3/ 


2.705 


5-046 


3.246 


5.887 


3-787 


6.728 


4.328 


7-569 


4.869 


VL* 


33° 


2.723 


5-032 


3.268 


5-871 


3-812 


6.709 


4-357 


7-548 


4.902 


sr 


33X 


2.741 


5.018 


3.290 


5.854 


3.838 


6.690 


4-386 


7-527 


4-935 


5634- 


33^ 


2.760 


5-003 


3-312 


5-837 


3.864 


6.671 


4.416 


7-505 


4.967 


56^ 


333/ 


2.778 


4.989 


3-333 


5.820 


3-88 9 


6.652 


4-445 


7-483 


5.000 


56/ 


34 u 


2.796 


4-974 


3-355 


5.803 


3-9 J 4 


6.632 


4-474 


7.461 


5-033 


56° 


34 1 / 


2.814 


4.960 


3-377 


5.786 


3-94o 


6.613 


4- 502 


7-439 


5-065 


55^ 


3VA 


2.832 


4-945 


3-398 


5.769 


3-965 


6-593 


4-531 


7-417 


5-098 


55^ 


34^ 
35° 


2.850 


4,930 


3.420 


5-752 


3-99° 


6-573 


4.560 


7-395 


5-130 


55X 

55° 


2.868 


4-9 J 5 


3-441 


5-734 


4.015 


6-553 


4-5S9 


7-372 


5.162 


35^ 


2.886 


4.900 


3-463 


5-7i6 


4.040 


6-533 


4.617 


7- 3 5o 


5-194 


543/ 


35K 


2.904 


4-885 


3-484 


5-699 


4.065 


6.513 


4.646 


7-327 


5.226 


54K 


353/ 


2.921 


4.869 


3-505 


5-681 


4.090 


6-493 


4.674 


7-304 


5-25S 


54# 


36 c 


2-939 


4-854 


3-527 


5.663 


4-H5 


6.472 


4.702 


7 .2Sr 


5-290 


54 


36/ 


2-957 


4-839 


3-548 


5-645 


4-139 


6.452 


4-730 


7.258 


5-322 


53 H 


36K 


2.974 


4.823 


3-569 


5.627 


4.164 


6.431 


4-759 


7-235 


5-353 


S3Vz 


363/ 


2.992 


4.808 


3-59° 


5-609 


4.1S8 


6.410 


4.7S7 


7.211 


5-3S5 


53# 


3r 


3.009 


4.792 


3.611 


5-590 


4.213 


6.389 


4.S15 


7.1SS 


5.416 


53° 


37^ 


3.026 


4.776 


3-632 


5-572 


4-237 


6.368 


4.S42 


7.164 


5-44S 


52 3/ 


37^ 


3-044 


4.760 


3-653 


5-554 


4.261 


6-347 


4.S70 


7.140 


5-479 


5- l 2 


373/ 


3.061 


4-744 


3-673 


5-535 


4.286 


6.326 


4.S9S 


7.116 


5-5io 


52# 


38° 


3.078 


4.728 


3-694 


5-5i6 


4.310 


6.304 


4-925 


7.092 


5-541 


52° 


38/ 


3-095 


4.712 


3-715 


5-497 


4-334 


6.2S3 


4-953 


7.06S 


5-572 


51 X 


38^ 


3-H3 


4.696 


3-735 


5.47S 


4-35S 


6.261 


4.980 


7-043 


5.603 


51 I, 


3 , 8 i< 


3-I30 


4.679 


3-756 


5-459 


4.381 


6.239 


5.007 


7019 


5.633 


5i \ 


39° 

CO 

CD 


3-147 


4.663 


3-776 


5-44o 


4-405 


6.217 


5-035 


6.904 


5.604 


51 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. | Lat. 


Dep. 


Lat. 


•cub 

Lea 1 


5 


6 


7 


8 


9 



48 



LATITUDES AND DEPARTURES. 



DO 

CO 

5" 


1 


2 


3 


4 


5 


.2 

cti 
OQ 

51 e 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


39 


0.777 


O.629 


J-554 


I.259 


2.331 


i.SSS 


3-109 


2.517 


3.SS6 


39* 


0.774 


O.633 


1-549 


1.265 


2.323 


I.89S 


3-098 


2.531 


3-872 


503/ 


29/ 2 


0.772 


O.636 


! 1-543 


I.272 


2.315 


1. 90S 


i 3.086 


2-544 


3.S5S 


50^ 


39^: 


O.769 


0.639 


! 1-538 


T.279 


2.307 


1. 918 


i 3-075 


2.55S 


3-344 
3-S30 


50* 


40 J 


O.766 


O.643 


1-532 


I.2S6 


2.298 


I.92S 


3.064 


2.571 


50° 


40 u 


O.763 


0.646 


1.526 


I.292 


2.290 


1.938 


3-o53 


2.5S4 


3.816 


49* 


AO l / 2 


O.760 


O.649 


1. 521 


I.299 


2.281 


1.948 


3.042 


2.598 


3.802 


49 Y2 


40* 


O.75S 


O.653 


I-5I5 


I.306 


2.273 


1.958 


3-030 


2. 6ll 


3.788 


49. X 


41° 


0.755 


O.656 


1.509 


1. 312 


2.264 


I.96S 


3.019 


2.624 


3-774 


49 


41* 


0.752 


O.659 


1-504 


I-3I9 


2.256 


1.978 


3.007 


2.637 


3-759 


483/ 


4*Y 


O.749 


O.663 


1.49S 


I.325 


2.247 


I.9SS 


2.996 


2.65O 


3-745 


48^ 


41* 
42 


O.746 


O.666 


1.492 


1-332 


2.238 


1.998 


2.9S4 


2.664 


3-73Q 


48* 


0.743 


O.669 


1.486 


1.338 


2.229 


2.007 


2-973 


2.677 


3.716 


48^ 


42* 


O.740 


0.672 


1.4S0 


1-345 


2.221 


2.017 


2.961 


2.6S9 


3- 7oi 


47* 


42 Yz 


0-737 


O.676 


1-475 


1. 351 


2.212 


2.027 


2.949 


2.702 


3.686 


47K 


42 3/ 


0-734 


O.679 


1.469 


1.353 


2.203 


2.036 


2-937 


2./I5 


3.672 


47^ 


43 3 


0.731 


O.6S2 


1.463 


I064 


2.194 


2.046 


2.925 


2.728 


3-657 


4V 


43 X 


O.728 


O.685 


1-457 


I-370 


2.IS5 


2.056 


2.913 


2.741 


3.642 


46* 


43 % 


0.725 


0.6S8 


I-45I 


1-377 


2.176 


2.065 


2.901 


2-753 


3-627 


46* 


43* 


O.722 


0.692 


1-445 


1-383 


2.167 


2.075 


2.8S9 


2.766 


3.612 


46^ 


44 


O.719 


0.695 


1-439 


1.3S9 


2.158 


2.0S4 


2.877 


2.779 


3-597 


46° 


44% 


O.716 


0.698 


1-433 


1.396 


2.149 


2.093 


2.865 


2.79I 


3.582 


45* 


44 Yz 


0.713 


0.701 


1.427 


1.402 


2.I4O 


2.103 


2.853 


2.804 


3-566 


45^2 


44* 


O.710 


0.704 


1.420 


T.408 


2.I3I 


2. 112 


2.841 


2.8l6 


3-551 


45* 


45 J 


0.707 


0.707 


1.414 


1-414 


2. 121 


2.T2I 


2.S28 


2.S28 


3-536 


45 ; 


B'riDg 


Dep. 


Lat. 


| Dep. 


Lat. 


; Dep. 


Lat. 


: Dep. 


Lat. 


| Dep. 


B'ring 


CO 

CD 

5. 

q©. 


3 


6 


7 


8 


9 


pa 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


39° 


3-147 


4.663 


3-776 


5 -440 


4-405 


6.217 


5-035 


6.994 


5-664 


51° 


39* 


3 


164 


4.646 


3-796 


5 


421 


4.429 


6.I95 


5.062 


6.970 


5-694 


5o* 


39^ 


3 


180 


4.630 


3.816 


5 


401 


4-453 


6.173 


50S9 


6-945 


5-725 


50* 


39* 


3 


197 


4.613 


3-837 


5 


382 


4.476 


6.151 


5. Il6 


6.920 


5-755 


50* 


40° 


3 


214 


4-59 6 


3-857 


5 


362 


4-5oo 


6.I2S 


5.I42 


6.S94 


5-7S5 


50 


40* 


3 


231 


4-579 


3-877 


5 


343 


4-523 


6.IO6 


5.169 


6.869 


5-8i5 


49* 


40K 


3 


247 


4-562 


3-897 


5 


323 


4-546 


6.0S3 


5.196 


6.844 


5-845 


49 yi 


4">* 


3 


264 


4-545 


3-9!7 


5 


303 


4569 


6.061 


5.222 


6.81S 


5-875 


49* 


4r 


3 


2 SO 


4-528 


3-936 


5 


2S3 


4-592 


6.038 


5.24S 


6.792 


5-9°5 


49 


41* 


3 


297 


4-5U 


3-956 


5 


263 


4.615 


6.015 


5-275 


6.767 


5-934 


48* 


41K 


3 


313 


4-494 


3.976 


5 


243 


4-63S 


5-992 


5-301 


6.741 


5-964 


43/ 2 


4itf 


3 


329 


4.476 


3-995 


5 


222 


4.661 


5-968 


5-327 


6.715 


5-993 


48* 


42 


3 


346 


4-459 


4.015 


5 


202 


4.6S4 


5-945 


5-353 


6.6SS 


6.022 


48 


42* 


3 


362 


4.441 


4-034 


5 


1S2 


4.707 


5.922 


5-379 


6.662 


6.051 


47* 


42^ 


3 


373 


4.424 


4-054 


5 


161 


4.729 


5.S9S 


5-405 


6.635 


6.0S0 


47^ 


42 3/ 


3 


394 


4.406 


4-073 


5 


140 


4-752 


5-875 


5-430 


6.609 


6.109 


47J4 


43' 


3 


410 


4-353 


4.092 


5 


119 


4-774 


5.851 


5-456 


6.5S2 


6.138 


4V 


43* 


-* 


426 


4-37Q 


4. in 


5 


099 


4.796 


5-827 


5.481 


6-555 


6.167 


4634 


43K 


3 


442 


4-352 


4.130 


5 


07S 


4.818 


5.803 


5-507 


6528 


6.195 


46* 


43* 


3 


45S 


4o34 


4.149 


5 


057 


4,841 


5-779 


5-532 


6.501 


6.224 


46J4 


44 


3 


473 


4.316 


4. 16S 


5 


035 


4.S63 


5-755 


5 557 


6.474 


6.252 


46 


44* 


; 


4S9 


4.295 


4.1S7 


5 


014 


4SS5 


5-730 


5-582 


6-447 


6.280 


45* 


44^ 


-> 


505 


4.280 


4.206 


4 


993 


4.906 


5.706 


5.607 


6.419 


6.308 


45^ 


44 * 


3 


520 


4.261 


4.224 


4 


97i 


4928 


5 6S1 


5.632 


6.392 


6.336 


45* 


45 


3 


536 


4-243 


4-243 


4 950 


4.950 


5-657 


5-657 


6.364 


6.364 


45" 


Bung 


Lat. 


Dep. 


Lat. 


Ppn 


Lat 


Dep. 


Lat. 


Dep 


Lat. 


B'riig: 



TABLES AND FORMULAS. 



49 



TABLES OF THE AVERAGE RESISTING VALUES 

OF MATERIALS COMMONLY USED IN 

ENGINEERING CONSTRUCTION. 



AVERAGE VALUES IN TENSION. 



Material. 



Coefficient of 
Elasticity. 

E 1 . 



Elastic 
Limit. 



Ultimate 

Tensile 

Strength. 

Si. 



Ultimate 
Elonga- 



tion. 

Si. 



Timber 

Cast Iron 

Wrought Iron 
Steel 



Lb. per Sq. In. 
1,500,000 

15,000,000 
25,000,000 
30,000,000 



Lb. per Sq. In 
3,000 
6,000 
25,000 
50,000 



^•PerSq.In.^Xk 



10,000 

20,000 

55,°°° 
IOO.OOO 



O.OI5 
O.OO5 
o. 20 

O. IO 



AVERAGE VALUES IN COMPRESSION. 



Material. 


Coefficient of 
Elasticity. 

E,. 


Elastic 
Limit. 

Z 2 . 


Ultimate 

Compressive 

Strength. 

s,. 


Timber 

Brick 


Lb. per Sq. In. 
1,500,000 

6,000,000 
15,000,000 
25,000,000 
30,000,000 


Lb. per Sq. In. 
3,000 

25,000 
50,000 


Lb. per Sq. In. 

8,000 
2,500 

6, 000 


Stone 


Cast Iron 


90,000 

55,000 

1 50,000 


Wrought Iron 

Steel 



50 



TABLES AND FORMULAS. 



AVERAGE VALUES IN SHEAR. 



Material. 


Coefficient of 
Elasticity. 


Ultimate 
Shearing 
Strength. 

Ss. 


Timber (across the grain) 

Timber (with the grain) 

Cast Iron 

Wrought Iron 


400,000 

6,000,000 

15,000,000 


3,000 

600 

20,000 

50,000 

70,000 


Steel 



ULTIMATE STRENGTH IN FLEXURE. 



Material. 



Ultimate Strength of 
Flexure in Lb. per Sq. In. 

S 4 . 



Cast Iron .... 
Wrought Iron 

Steel 

Brass 

Ash 

Brick 

Stone 

Hemlock 

Oak, white. . . 
Pine, w T hite . . . 
Pine, yellow. . 
Hickory , 



45,000 

20,000 

17,000 

14,000 

1,000 

2,000 

7,200 

12,500 

9,000 

1 1,000 

16,000 



FACTORS OF SAFETY TO BE USED IN CONNECTION 
WITH THE FOUR PRECEDING TABLES. 



Material. 


For Steady 

Stress. 
(Buildings.) 


For Varying 

Stress. 
(Bridges.) 


For Shocks. 
(Machines.) 


Timber 

Brick and Stone . . . 

Cast Iron 

Wrought Iron 

Steel 


8 

6 
4 

5 


10 

25 

10 

6 


*5 

3° 

T 5 
10 
10 







TABLES AND FORMULAS. 51 



TABLE OF 

HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS. 

The formulas used in the computation of the following 
tables furnish expressions for horizontal distances and 
differences of elevation for stadia measurements with the 
conditions that the stadia rod be held vertical and the stadia 
wires be equidistant from the center wire. The formulas 
used are as follows: For the horizontal distance 

D = c cos n + a k cos 2 », (94.) Art. 1 301 . 

in which D = the corrected distance ; c = the constant ; 
a k = the stadia distance, and n = the vertical angle. 

For the difference of elevation, the following formula is 
used: 

E = csin n + ak^J 1 — . (95.) Art. 1301. 

For application of tables see Art. 1301. 



TABLES AND FORMULAS. 



53 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS. 



Minutes. 



O ..... . 

2 

4 

6 ... .., 

8 

io 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

30 

38 

40 

42 ...... 

44 

46 

48 

50 

52 

54 

56 

58 

60 

c= .75 

C = I. OO 

c = 1.25 



Hor. 
Dist. 

100.00 
100.00 
100.00 
100.00 
100.00 
100.00 

100.00 
100.00 
100.00 
100.00 
100.00 

100.00 
100.00 

99-99 
99-99 
99-99 

99-99 
99-99 
99-99 
99-99 
99-99 

99-99 
99.98 
99.98 
99.98 
99.98 

99.9S 
99.98 
99-97 
99-97 
99-97 



75 



1.25 



DIff. 
Elev. 



Hor. 
Dist. 

99-97 
99-97 
99-97 
99. 96 
99.96 
99.96 

99.96 
99-95 
99-95 
99-95 
99-95 

99.94 
99-94 
99-94 
99-93 
99-93 

99-93 
99-93 
99.92 
99.92 
99.92 

99.91 
99.91 
99.90 
99.90 
99.90 

99- §9 
99.89 
99.89 
99.88 
99.88 



75 



Diff. 
Elev. 

1.74 
1.80 
1.86 
1.92 
1.98 
2.04 

2.09 
2.15 
2.21 
2.27 
2.33 
2.38 
2.44 
2.50 
2.56 
2.62 

2.67 

2.73 
2.79 
2.85 
2.91 

2.97 
3.02 

3.08 

3-M 
3.20 

3.26 

3-3i 
3-37 
3-43 
3-49 



.02 
■03 



Hor. 
Dist. 

99.88 
99.87 
99.87 
99.87 

99. 86 
99.86 

99-85 
99-85 
99.84 
99.84 
99-83 

99-83 
99.82 
99.82 
99.81 
99.81 

99.80 
99.80 
99- 79 
99-79 
99.78 

99.78 
99-77 
99-77 
99.76 
99.76 

99-75 
99- 74 
99-74 
99- 73 
99-73 



•75 



Diff. 
Elev. 

3-49 



03 



"4 



05 



Hor. 
Dist. 

99-73 
99.72 
99.71 
99.71 
99- 7o 
99.69 

99.69 
99.68 
99.68 
99.67 
99.66 

99.66 

99-65 
99.64 

99-63 
99-63 
99.62 
99.62 
99.61 
99.60 
99-59 

99-59 
99-58 
99-57 
99- 56 
99- 56 

99-55 
99- 54 
99-53 
99.52 
99. 51 



75 



r.25 



Diff. 
Elev. 

5-23 

5.28 

5-34 
5-40 
5-46 

5-52 

5-57 
5-63 
5-69 

5-75 
5- 80 

5-86 
5-9 2 
5-98 
6.04 
6.09 

6.15 
6.21 
6.27 

6-33 
6.38 

6.44 

6.50 
6.56 
6.61 
6.67 

6-73 
6.78 
6.S4 
6.90 
6.96 



•05 



.06 
.08 



54 



TABLES AND FORMULAS. 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS* 



Minutes. 



O 

2 

4 

6 

8 

io 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36.... , 

38 

40 

42 

44 

46 

48 

5o 

52 

54 

56 

58 

60 

c = -75 
c = 1. 00 
c = 1.25 



Hor. 
Dist. 

99-51 
99-51 
99-5Q 
99-49 
99-48 
99-47 

99.46 
99.46 
99-45 
99-44 
99-43 

99.42 
99.41 
99.40 
99-39 
99-38 

99-38 
99-37 
99-36 
99-35 
99-34 

99-33 
99-32 
99-31 
99-3o 
99.29 

99. 2S 
99.27 
99.26 
99-2 5 
99.24 



75 



1.25 



Diff. 
Elev. 

6.96 
7.02 
7.07 

7-13 
7.19 

7-25 

7-3o 
7-36 
7.42 
7.4S 

7-53 

7-59 
7.65 
7.71 
7.76 

7.S2 

7. 88 
7-94 
7-99 
S.05 
S.11 

S.17 
8.22 
S.2S 

8.34 
8.40 

S-45 
S.51 
S-57 
8.63 
8.68 



.06 



Hor. 
Dist. 

99.24 

99- 2 3 
99.22 
99.21 
99.20 
99.19 

99.18 
99.17 
99.16 

99-15 
99.14 

99-13 
99.11 
99.10 
99.09 
99.08 

99.07 
99.06 

99-°5 
99.04 

99-03 

99.01 
99.00 
98.99 
98.98 
9S.97 

98.96 
98.94 

93-93 
98.92 
98.91 



75 



99 



1.24 



Diff. 
Elev. 

8.68 
S.74 
8.80 
8.85 
S.91 
S.97 

9- °3 
9.08 
9.14 
9.20 
9-25 

9-3i 
9-37 
9-43 
9.48 

9-54 
9.60 

9-65 
9.71 

9-77 

9-33 

9.88 

9.94 

10.00 

10.05 

10. 11 

10.17 
10.22 
10.28 
10.34 
10.40 



■07 



.09 



Hor. 
Dist. 

98.91 
98.90 
98.88 
9S.87 
98.86 
9S.S5 

98.83 
98.82 

98. Si 
98.80 

9S.7S 

9S.77 
9S.76 
98.74 
9S.73 
98.72 

9S.71 
98.69 
98.68 
9S.67 
9S.65 

9S.64 
98-63 
98.61 

98. 60 
98.58 

98.57 
9S.56 
9S.54 
98.53 
98.51 



75 



•99 



1.24 



Diff. 
Elev. 

0.40 
0.45 
0.51 
0.57 
0.62 
0.68 

0.74 
0.79 
0.S5 
0.91 
0.96 

1.02 
1.08 

i- 13 

1. 19 

1.25 

1.30 
1.36 
1.42 
1.47 

i-53 

i-59 
1.64 
1.70 
1.76 
1. Si 

1.87 

i-93 
1.98 
2.04 
2.10 



.08 



M 



7 



Hor. 
Dist. 

9S.51 
98.50 
9S.4S 
98-47 
98.46 
98.44 

98-43 
9S.41 
98.40 
98-39 
9S.37 

98-36 
98-34 
98-33 
98.31 
98.29 

93.28 
98.27 
98.25 
98.24 
98.22 

98.20 
98.19 
98.17 
98.16 
9S.14 

98-13 
98.11 
98. 10 
98.08 

08.06 



74 



■ 99 



1.24 



Diff. 

Elev. 

12.10 

12.15 

12.21 

12.26 

12.32 

12.38 

12.43 
12.49 

12.55 
12.60 
12.66 

12.72 
12.77 
12.83 
12.88 
12.94 

13.00 
13-05 
13.11 
13-17 
13.22 

13-28 
13-33 

13-39 
13-45 

i3-5o 

I3-56 
13 61 
I3-67 
13-73 
I3-78 



TABLES AND FORMULAS. 



oo 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS. 



Minutes. 



o 

2 

4 

6 

8 

io 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 ..... . 

46 

48 

50 

52 

54 

56 

58 

60 

c= .75 

C — I. OO 
C = 1.25 



Hor. 

Dist. 

98.06 
98.05 
98.03 
98.01 
98.00 
97.98 

97-97 
97-95 
97-93 
97.92 
97.90 

97.88 
97.87 
97.85 
97-83 
97.82 

97.80 
97.78 
97.76 
97-75 
97- 73 

97-71 
97.69 
97.68 
97.66 
97.64 

97.62 
97.61 
97-59 
97-57 
97-55 



74 



.99 



Diff. 
Elev. 

13-78 
13 
13 
13 

14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 

14 

14, 

14 
14 
14 
14 

14 

15 

15 
15 

15 

15 
15 
15 
15 
15 



1-23 



i? 



iS 



Hor. 
Dist. 

97-55 
97-53 
97-52 
97-5Q 
97.48 
97.46 

97-44 
97-43 
97.41 

97-39 
97-37 

97-35 
97-33 
97-31 
97.29 
97.28 

97.26 
97.24 
97.22 
97.20 
97.18 

97.16 
97.14 
97.12 
97.10 
97.08 

97.06 
97.04 
97.02 
97.00 
96.98 



74 



•99 



Diff. 
Elev. 

15-45 
15-51 
15-56 
15.62 
15-67 

15-73 

15-78 
15.84 
15.89 

15-95 
16.00 

16.06 
16.11 
16.17 
16.22 

16.28 

16.33 
16.39 
16.44 
16.50 
16.55 

16.61 
16.66 
16.72 

16.77 
16.83 

16.88 
16.94 
16.99 
17-05 
17.10 



16 



Hor. 
Dist. 

96.98 
96.96 
96.94 
96.92 
96.90 
96.88 

96.86 
96.84 
96.82 
96.80 
96.78 

96.76 
96.74 
96.72 
96.70 
96.68 

96.66 
96.64 
96.62 
96.60 
96-57 

96-55 
96-53 
96.51 

96-49 
96-47 

96-45 
96.42 
96.40 
96.38 
96.36 



Diff. 
Elev. 

7.10 
16 

21 



74 



, 9 S 



21 I.2T 



I 1 



iS 



23 



1 1 



Hor. 
Dist. 

96.36 
96 



Diff, 
Elev. 

" 73 
78 
84 



73 



98 



/1 



26 



56 



TABLES AND FORMULAS. 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS, 



Minutes. 



O 

2 

4 

6 

8 

io 

12 

M 

16 

18 ...... 

20 

22 

24 

26 

2S 

3D 

32 

34 

36.... . 

38 

40 

42 

44 

46 

4S 

50 

52 

54 

56...... 

58 

60 

C= -75 
C = I. OO 

c = 1.25 



Hor. 
Dist. 

95-6S 
95-65 
95-63 
95.61 
95-5S 
95-56 

95-53 
95-51 
95-49 
95-46 
95-44 

95-41 
95-39 
95-36 
95-34 
95-32 

95-29 
95-2? 
95-24 
95-22 

95-19 

95-17 
95-14 
95-12 
95-09 
95-Q7 

95-Q4 
95.02 

94-99 
94-97 
94-94 



.98 



Diff. 
Elev. 
20.34 
20.39 

20.44 
20.50 

20.55 
20.60 

20.66 

20.71 
20.76 

20. Si 
20.87 

20.92 
20.97 
21.03 
21. oS 
21.13 

21. iS 
21.24 
21.29 

21-34 
21.39 

21.45 
21.50 

21-55 
21.60 
21.66 

21.71 
21.76 
21. Si 
21.87 



1 3 



Hor. 
Dist. 

94-94 
94.91 
94. Sg 
94. 86 
94. S 4 
94. Si 

94-79 
94.76 

94-73 
94-71 
94. 6S 

94.66 
94-63 
94.60 
94-58 
94-55 

94-52 
94-5Q 
94-47 
94-44 
94.42 

94-39 
94-36 
94-34 
94-31 
94-23 

94.26 
94-23 
94.20 

94-17 



21.92 94.15 



16 I 



73 



■97 



Diff. 
Elev. 
21.92 

21.97 
22.02 
22.08 
22.13 

22.1S 

22. 23 
22.28 
22.34 
22.39 
22.44 
22.49 

22.54 
22.60 
22.65 
22.70 

22.75 
22.80 
22.85 
22.91 
22.96 

23.01 
23.06 
23.11 
23.16 
23.22 

23.27 
23-32 

23-37 
23.42 

23-47 



14 



Hor. 
Dist. 

94-15 
94.12 
94.09 
94.07 
94.04 
94.01 

93-9 s 
93-95 
93-93 
93-9° 
93-87 

93-84 
93.81 

93-79 
93-76 
93-73 

93-7Q 
93-67 
93-65 
93.62 

93-59 
93-56 
93-53 
93-5o 
93-47 
93-45 

93-42 
93-39 
93-36 
93-33 
93-3o 



73 



Diff. 
Elev. 

23-47 
23.52 
23-58 
23-63 
23.68 
23-73 

23-73 
23-83 

23.8S 

23-93 
23-99 
24.04 
24.09 
24.14 
24.19 
24.24 

24.29 
24-34 
24-39 
24.44 
24.49 

24-55 
24.60 
24.65 
24.70 

24-75 
24.80 

24-85 
24.90 

24-95 
25.00 



15 



■97 



Hor. 

Dist. 

93-30 
93.27 

93-24 
93.21 
93.18 
93.16 

93-13 
93.10 

93-07 
93-04 
93.01 

92.98 

92-95 
92.92 
92.89 
92.86 

92.83 
92.80 
92.77 

92-74 
92.71 

92.68 
92.65 
92.62 

92-59 
92.56 

92-53 
92-49 
92.46 

92-43 
92.40 



96 



3i 



Diff. 
Elev. 

25.00 

25-05 
25.10 

25-15 
25.20 

25-25 
25-30 

25-35 
25.40 

25-45 
25-50 

25-55 
25.60 
25-65 
25.70 

25-75 
25.80 
25-85 
25.90 
25-95 
26.00 

26.05 
26.10 
26.15 
26.20 
26.25 

26.30 
26.35 
26.40 

26.45 
26.50 



>27 



34 



TABLES AND FORMULAS. 



57 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS. 



Minutes. 



o 

2 

4 

6 

S 

io 

12 

14 
16 

18 

20 

22 

24 
26 
28 
30 

32 

34 
36 
33 
40 



42 
44 
46 
48 
5o 

52 
54 
56 
53 
60 



i6 ( 



c= .75 

C = I. OO 

c = 1.25 



Hor. 
Dist. 

92.40 

9 2 -37 
92.34 
92.31 
92.28 
92.25 

92.22 
92.19 

92-15 
92.12 
92.09 

92.06 
92.03 
92.00 
91.97 
9 x -93 
91.90 
91.87 
91.84 
91.81 
91.77 

9 x -74 
91.71 
91.68 
91.65 
91.61 

91.58 

9 x -55 
91.52 
91.48 
9 r -45 

•72 
.86 



DIff. 

Elev. 

26.50 

26.55 
26.59- 
26.64 
26.69 
26.74 

26.79 
26.84 
26.89 
26.94 
26.99 

27.04 
27.09 

27-13 
27.18 
27.23 

27.28 

27-33 
27.38 
27-43 

27.48 

27.52 

27-57 
27.62 

27.67 

27.72 

27.77 
27.81 
27.86 
27.91 
27.96 



2S 



i7 



35 



Hor. 

Dist. 

91-45 
91.42 

9*-39 
91-35 
91.32 
91.29 

91.26 
91.22 
91.19 
91.16 
91.12 

91.09 
91.06 
91.02 
90.99 
90.96 

90.92 
90.89 
90.86 
96.82 
90.79 

90.76 
90.72 
90.69 
90.66 
90.62 

90-59 
90-55 
90.52 
90.48 
90.45 

.72 

•95 

1. 19 



Diff. 

Elev. 

27.96 
28.01 
28.06 
28.10 
28.15 
28.20 

28.25 
28.30 
28.34 
2S.39 
28.44 
28.49 
2S.54 
28.58 
28.63 
28.68 

28. 73 

2S.77 
28.82 
28.S7 
28.92 

28.96 

29.01 
29.06 
29.11 

29-15 

29.20 

29.25 
29.30 

29-34 
29.39 

•23 

• 30 



Hor. 
Dist. 

90.45 
90.42 
90.38 

9°-35 
90.31 
90.28 

90.24 
90.21 
90.18 
90.14 
90. ir 

90.07 
90.04 
90.00 
89.97 
89-93 

89.90 
89.86 

89-83 
89.79 
89.76 

89.72 
89.69 
89.65 
89.61 

89.53 

Sg-54 

89.51 
S9.47 
89.44 
89.40 

• 71 

•95 



.38 j 1.19 



Diff. 
Elev. 

29-39 
29.44 
29.43 
29-53 
29.58 
29.62 

29.67 
29.72 
29.76 
29.81 
29.86 

29.90 

29-95 
30.00 
30.04 
30.09 

30.14 
30.19 
30.23 

30. 2S 
30.32 

30.37 
3O.4I 
30.46 
30.5I 

30.55 

3O.60 

30.65 
30. 69 

30. 74 

30.78 

.24 
•32 

.40 



10 



Hor. 
Dist. 

89.40 
89-36 
89-33 
89.29 
89.26 
89.22 

89.18 
89.15 
89. ir 
89.0s 
89.04 

89.00 
8S.96 

88.93 
88.89 
88.86 

88.82 
8S.78 

88.75 
8S.71 
SS.67 

SS.64 
SS.60 
SS.56 

SS.53 
SS.49 

SS.45 
8S.41 
SS.3S 

88.34 
88.30 

■7i 

.04 



Diff. 
Elev. 

30.78 
30.83 
30.87 
30.92 
30.97 
31.01 

31.06 
31.10 
31-15 
31-19 
31-24 

31-28 

31-33 
31-33 
31.42 
31-47 

3i-5r 
3I-56 
31.60 

31-65 
31.69 

31-74 
31.78 
31.83 
31-87 

31.92 

31.96 
32.01 
32.05 
32.09 
32.14 



58 



TABLES AND FORMULAS. 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS. 



o . 

2 . 

4 • 
6 . 

5 . 
io . 

12 . 

14 • 
16 . 
iS , 

20 . 

22 , 

24 
26 
2S 
30 

32 
34 
36 
33 
40 

42 

44 
46 
4 S 
5o 

52 
54 
56 
5S 
60 



Minutes. 



C= .75 
C = 1. 00 
c = 1.25 



21 



Hor. 
Dist. 

ss.30 
ss.26 
ss.23 
ss.19 
88.15 

S3.ii 

SS.oS 
88. 04 
8S.00 
37.96 
37-93 
5 7 .3 9 
87.85 
87.81 
37-77 
57.74 

S7.70 
S7.66 
S7.62 
87-58 

37-54 

37-51 
37.47 
37-43 
S7-39 
S7-35 

37-31 
37-27 
S7.24 
S7.20 
S7.16 



Diff. 
Elev. 

32.14 

32.18 

32.23 
32.27 
32.32 
32.36 

32.41 
32-45 
32-49 
32-54 

32-53 

32.63 
32.67 
32.72 
32.76 

32. So 

32.S5 
32.59 

32.93 
32.93 
33-Q2 

33-Q7 
33-n 
33-15 
33-20 
33-24 
33-23 
33-33 
33-37 
33-41 
33-46 



2t 



Hor. 
Dist. 

S7.16 
S7.12 
87.08 
37-04 
37.00 

36. 96 

36. 92 
S6.S3 
S6.54 
36. So 
S6.77 

S6.73 
36. 69 
S6.65 
86.61 
S6.57 

36. 53 
S6.49 
S6.45 
56.41 

S6.37 

36-33 
S6.29 
36.25 
S6.21 
S6.17 

S6.13 
S6.09 
S6.05 
S6.01 
S5-97 



:■-■ 



94 



•93 



1. 17 



.44 1. 16 



Diff. 
Elev. 

33-46 
33-50 

33-54 
33-59 
33-63 
33-67 

33- 72 
33-76 
33- So 
33-34 
33-39 

33-93 
33-97 
34-oi 
34.06 
34-io 

34-14 
34- iS 
34-23 
34-27 

34-31 

34-35 
34-4Q 
34-44 
34-43 
34-52 

34-57 
34.61 

34-65 
34-69 
34-73 



23 



Hor. 
Dist. 

S5-97 
35-93 
35-39 
B5.85 
S5.S0 
S5.76 

S5.72 
I 85.68 
S5.64 
S5.60 
35-56 

35-52 
S5.4S 
; 5-44 
55-40 
55.36 

35-31 
85.27 

35-23 
35-19 

35-15 

S5.11 
85-07 
S5.02 

84.98 

34-94 

S4.90 

S4.S6 
S4.S2 
34-77 

34-73 



.69 



• 12 



46. I.I5 



Diff. 
Elev. 

34-73 
34-77 
34-32 
34. S6 

34-9° 
34-94 

34-9S 
35.02 
35-07 
35-n 

35-15 

35-19 
35-23 
35-27 
35-31 
35-36 

35-40 
35-44 
35-43 
35-52 
35-56 

35-6o 

35-64 
35-65 
35-72 
35-76 

35-So 

35.S9 
35-93 

35-97 



29 



Hor. 

Dist. 

34-73 
S4.69 
34.65 
S4.61 
34-57 
84-52 

34.43 
54.44 
34-40 
S4-35 
34-31 

54.27 
34-23 
34.13 

34-14 
54.10 

S4.06 
34.01 
S3-97 
33-93 

S3.S9 

53.54 
S3. 80 
33-76 
33-72 
83.67 

33-63 
33-59 
33-54 
S3.50 

53.46 



.69 



92 



Diff. 
Elev. 

35-97 
36.01 
36.05 
36.09 
36.13 
36.17 

36.21 
36.25 
36.29 
36.33 
36.37 

36.41 
36.45 
36.49 
36.53 
36.57 

36.61 
36.65 
36-69 
36.73 
36.77 

36. So 
36. S 4 
36.SS 
36.92 
36.96 

37.00 
37-Q4 
37-oS 
37.12 
37-i6 



30 



.40 



.48 



50 



TABLES AND FORMULAS. 59 

HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS. 



Minutes. 



O 

2 

4 

6 

S 

io 

12 

14 

16 

iS 

20 

22 

24 

26 

28 

SO 

32 

34 

36 

38 

40 

42 

44 

4& 

48 

5o 

52 

54 

56 

58 

60 

C= .75 

c = I. OO 

c = 1.25 



24 



Hor. 
Dist. 

83.46 
83.41 

S3-37 

83-33 
83.28 
S3.24 

83.20 

83-15 
83.11 
83.07 
83.02 

82.98 
82.93 
82.89 
82.85 

82.80 

82.76 
82.72 
82.67 
82.63 
82.58 

82.54 
82.49 

82.45 
82.41 
82.36 

82.32 
82.27 
82.23 
82.18 
82.14 



.68 



.91 



1. 14 



Diff. 
Elev. 

3716 



41 



52 



2 5 



Hor. 
Dist. 

82.14 
S2.09 
82.05 
82.01 
81.96 
81.92 

81.87 
81.83 
81.78 
81.74 
81.69 

81.65 
81.60 
81.56 
81.51 
81.47 

81.42 
81.38 

81.33 
81.28 
81.24 

Si. 19 
81.15 
81.10 
81.06 
81.01 

80.97 
80.92 
80.87 
80. S3 
80. 7S 



.68 



.90 



I-I3 



Diff. 
Elev. 

38.30 



43 



54 



2 6 ( 



Hor. 
Dist. 

80.78 
' 74 
69 
65 
60 
55 



Diff. 

Elev. 

39-40 

39-44 
39-47 
39-51 
39-54 
39-58 

39.61 
39-65 
39-69 
39-72 
39-76 

39-79 
39-83 
39.86 

39-9° 
39-93 

39-97 
40.00 
40.04 
40.07 
40.11 

40.14 
40.18 
40.21 
40.24 
40. 2 S 

40.31 
40.35 
40.3S 
40.42 
40.45 



27 



Hor. 
Dist. 

79.39 

79-34 
79-30 
79-25 
79.20 

79.15 

79.11 
79.06 
79.01 
78.96 
78.92 

78.87 
78.82 
78.77 
78.73 
78.68 

78.63 
7S.58 
78.54 
7S.49 
78.44 

7S.39 
78.34 
7S.30 
78.25 
7S.20 

7S.I5 
7S.10 
7S.06 
7S.01 

77.06 



66 



56 



Diff. 
Elev, 

40.45 
40.49 
40.52 
40-55 
40.59 
40.62 

40.65 
40.69 
40.72 
40.76 
40.79 
40.82 
40.86 
40.89 
40.92 
40.96 

40.99 
41.02 
41.06 
41.09 
41.12 

41.16 
41.19 
41.22 
41.26 
41.29 

41-32 
41-35 

41-39 
41.42 

41-45 



46 

.t:8 



6<; 



TABLES AND FORMULAS. 



HORIZONTAL DISTANCES AND DIFFERENCES 
OF ELEVATION FOR STADIA MEASUREMENTS, 



12 . 

14 ■ 
16 . 

iS . 
.20 , 

22 

24 
26 
20 
30 

32 

34 
36 
33 
-P 
42 

44 
46 

4 S 
5o 

52 
54 
56 
58 
60 



Minutes. 



C = -75 
C = I. OO 

c = 1.25 



Hor, 
Dist. 

77.96 

77-9 1 
77-86 

77.81 

77-77 
77.72 

77.67 
77.62 

77-57 
77-52 
77.4S 

77.42 

77-38 
77-33 
77.2S 

77-23 
77.18 

77-13 
77.09 
77.04 
76.99 

76.94 
76. 89 
76. S 4 
76.79 
76.74 

76.69 
76.64 
76.59 
76.55 
76.50 



.66 



.88 



Diff. 
Elev. 

41.45 
41.48 
41-52 
41-55 
41.5s 
41.61 

41.65 
41.68 

41.71 
41.74 

41-77 

41.81 
41.84 

41-87 
41.90 

41-93 

41.97 
42.00 
42.03 
42.06 
42.09 

42.12 
42.15 
42.19 
42.22 

42.25 

42.28 
42.31 
42.34 
42.37 
42.40 



29 



,36 



Hor. 
Dist. 

76.50 

76.45 
76.40 

76.35 
76.30 
76.25 

76.20 

76.15 
76.10 
76.05 
76.00 

75-95 
75-9° 

75-S5 
75.80 

75-75 

75-7Q 
75-65 
75.60 

75-55 
75-5Q 

75-45 
75.40 
75-35 
75-30 
75-25 
75.20 

75-15 
75-io 

75-05 
75.00 



Diff. 


Hor. 


Elev. 


Dist. 


42.40 


75.00 


42.43 


74-95 


42.46 


74.90 


42.49 


74- S5 


42.53 


74. So 


42.56 


74-75 



.4- 



, 6( > 



,65 



.87 



1.09 



42.59 
42.62 

42.65 
42. 6S 
42.71 

42.74 
42.77 
42.80 
42.83 
42.86 

42. S 9 
42.92 

42.95 

42.98 
43.01 

43-04 
43-07 
43.10 

43-13 
43.16 

43.18 

43.21 
43-24 
43-27 
43-3o 



3° 



37 



49 



.62 



74.70 

74-65 
74.60 

74-55 
74-49 

74-44 
74-39 
74-34 
74.29 
74.24 

74.19 

74-14 
74.09 
74.04 
73-99 

73-93 

73.88 

73.83 
73- 7S 
73-73 
73-6S 
73-63 
73-5S 
73-52 
73-47 



.86 



1.08 



Diff. 
Elev. 

43-30 

43-33 
43-36 
43-39 
43-42 
43-45 

43-47 
43-5o 
43-53 
43-56 
43-59 
43.62 
43-65 
43-67 
43-70 
43-73 

43-76 

43-79 

43.S2 

43-84 

43 S7 

43-90 
43-93 
43-95 
43-98 
44.01 

44.04 
44.07 
44.09 
44.12 
44-15 

~ 33 



■ 51 
.64 



TABLES AND FORMULAS. 61 



TABLE OF 

RADII AND CHORD AND TANGENT 
DEFLECTIONS. 

The formulas used in the computation of the following 
tables are as follows: 

For Radii, R = -4^. (89.) Art. 1249. 
sin V v ' 

For Chord Deflections, 

d=^- (92.) Art. 1255. 
For Tangent Deflections, 

tan deflection = — =. (93.) Art. 1255. 



TABLES AND FORMULAS. 63 

TABLE OF RADII AND DEFLECTIONS. 









Tan- 








Tan- 








Tan- 


De- 


Radii. 


Chord 
Deflec- 


gent 
De- 


De- 


Radii. 


Chord 
Deflec- 


gent 
De- 


De- 


Radii. 


Chord 
Deflec- 


gent 
De- 


gree. 




tion. 


flec- 
tion. 


gree. 




tion. 


flec- 
tion. 


gree. 




tion. 


flec- 
tion. 


o 5 


68754.94 


• 145 


•073 



5 15 


1091.73 


9. 160 


4.580 


/ 
10 50 


529.67 


18.880 


9.440 


IO 


34377-48 


.291 


• 145 


20 


1074.63 


9-305 


4 


653 










15 


22918.33 


•436 


.218 


25 


1058.16 


9-45o 


4 


725 


11 


521.67 


19.169 


9-585 


20 


17188.76 


.582 


.291 


30 


1042.14 


9-596 


4 


798 


10 


5 J 3-9i 


19-459 


9.729 


25 


13751.02 


•727 


•3 6 4 


35 


1026.60 


9.741 


4 


870 


20 


506.38 


19.748 


9.874 


30 


11459.19 


•873 


•436 


40 


ion. 51 


9.886 


4 


943 


30 


499.06 


20.038 


£0.019 


35 


9822.18 


1. 018 


•509 


45 


996.87 


10.031 


5 


<->ir, 


40 


491.96 


20.327 


10.164 


40 


8594.41 


1. 164 


•58- 


50 


9S2.64 


10.177 


5 


- ,88 


50 


485.05 


20.616 


10.308 


45 


7639-49 


1 . 309 


• 654 


55 


968.81 


10.322 


5 


161 










50 


6875-55 


1-454 


.727 












12 


478.34 


20.906 


io.453 


55 


6250.51 


1.600 


.80c 


6 


955-37 


10.467 


5 


234 


10 


471.81 


21.195 


io.597 










5 


942.29 


10.612 


5 


306 


23 


465.46 


21.484 


10.742 


1 


5729.65 


1-745 


.873 


10 


929-57 


10.758 


5 


37" 


30 


459.28 


21-773 


10.887 


5 


5288.92 


1. 891 


•945 


15 


917.19 


10.903 


5 


45i 


40 


453-26 


22 .063 


11. 031 


10 


4911.15 


2.036 


T.018 


20 


905-13 


11.048 


5 


524 


50 


447.40 


22 .352 


11. 176 


15 


4583-75 


2.182 


1. 091 


25 


893-39 


11. 193 


5 


597 










20 


4297.28 


2.327 


1. 164 


30 


881.95 


"•339 


5 


66g 


13 O 


441.68 


22.641 


11 .320 


25 


4044.51 


2.472 


1.236 


35 


870.79 


II.48^ 


5' 


742 


IO 


436.12 


22.930 


11.465 


30 


3819.83 


2.618 


1.309 


40 


859.92 


11.629 


5 


814 


20 


430.69 


23.219 


1 1 . 609 


35 


3618.80 


2.763 


1.382 


45 


849.32 


11.774 


5 


887 


30 


425.40 


23-507 


11 75+ 


40 


3437-87 


2.909 


1-454 


50 


838.97 


11. 910 


5 


960 


40 


420.23 


23.796 


11.898 


45 


3274-17 


3-o54 


1-527 


55 


828.88 


12.065 


6 


032 


50 


4i5-i9 


24.085 


12.043 


50 


3!25-3 6 


3.200 


1.600 




















55 


2989.48 


3-345 


1-673 


7 


819.02 


12.210 


6 


105 


14 O 


410.28 


24-374 


12.187 










5 


809 . 40 


12.355 


6 


177 


IO 


405-47 


24.663 


12. 33 1 


2 


2864.93 


3-49° 


1-745 


10 


800 . 00 


12.500 


6 


250 


20 


400.78 


24.951 


12.476 


5 


2750.35 


3-636 


1. 818 


15 


790.81 


12.645 


6 


323 


30 


396 . 20 


25.240 


12.620 


10 


2644.58 


3.781 


1. 891 


20 


781.84 


12.790 


6 


395 


40 


391.72 


25.528 


12.764 


15 


2546.64 


3-927 


1.963 


25 


773.07 


12.936 


6 


468 


50 


387-34 


25.817 


12.908 


20 


2455.70 


4.072 


2.036 


30 


764.49 


13.081 


6 


54" 










25 


2371.04 


4.218 


2. iog 


35 


756.10 


13.226 


6 


613 


15 O 


383.06 


26.105 


13-053 


3° 


2292.01 


4-363 


2. 181 


40 


747.89 


13-371 


6 


68 s 


IO 


378.88 


26.394 


I3.I97 


35 


2218.09 


4.508 


2.254 


45 


739.86 


i3-5i6 


6 


73S 


20 


374-79 


26.682 


13-341 


40 


2148.79 


4-654 


2.327 


50 


732.01 


13.661 


f: 


831 


30 


370.78 


26.970 


13-485 


45 


2083.68 


4-799 


2.400 


55 


724.31 


13.806 


6 


903 


40 


366.86 


27.258 


13-629 


50 


2022.41 


4-945 


2.472 












50 


363.02 


27-547 


13-773 


55 


1964.64 


5.090 


2-545 


8 


716.78 


i3-95i 


6 


976 


















5 


7°9 • 40 


14.096 


7 


,48 


16 


359.26 


27.835 


13-917 


3 


1910.08 


5-235 


2.618 


10 


702.18 


14.241 


7 


121 


IO 


355-59 


28.123 


14.061 


5 


1858.47 


5-38i 


2.690 


1 5 


695.09 


14-387 


7 


19 3 


20 


35I-98 


28.411 


14.205 


10 


1809.57 


5-526 


2.763 


20 


688.16 


14-532 


7 


266 


30 


348.45 


28.699 


14-349 


15 


1763.18 


5.672 


2.836 


25 


681.35 


14.677 


7 


.338 


40 


344-99 


28.986 


14-493 


20 


1719.12 


5-8i 7 


2.908 


30 


674 . 69 


14.822 


7 


411 


50 


341.60 


29.274 


14-637 


25 


1677.20 


5.962 


2.981 


35 


668.15 


14.967 


7 


|88 










30 


1637.28 


6.108 


3-054 


40 


661.74 


15. 112 


7 


^3<< 


17 


338.27 


29.562 


14.781 


35 


1599.21 


6.253 


3-!27 


45 


655-45 


15-257 


7 


628 


IO 


335-Qi 


29.850 


14.925 


40 


1562.88 


6.398 


3-199 


50 


649.27 


15.402 


7 


701 


20 


331.82 


30.137 


15.069 


45 


1528.16 


6-544 


3.272 


55 


643.22 


15-547 


7 


773 


30 


328.68 


30.425 


15.212 


5o 


I 494-95 


6.689 


3-345 












43 


325.60 


30.712 


I5-356 


55 


1463.16 


6.835 


3-4*7 


9 

5 


637.27 
631.44 


15.692 
15-837 


7 
7 


846 

918 


50 


322.59 


31 .000 


15-500 


4 


1432.69 


6.980 


3-490 


10 


625.71 


15-982 


7 


991 


18 


319.62 


31.2S7 


15-643 


5 


1403.46 


7-125 


3-563 


15 


620.09 


16.127 


8 


, .63 


IO 


316.71 


31-574 


15-787 


10 


I375-40 


7.271 


3 635 


20 


614.56 


16.272 


8 


136 


20 


313.86 


31.861 


I5-93I 


15 


1348.45 


7.416 


3.708 


25 


609 . 1 4 


16.417 


8 


208 


30 


311.06 


32.149 


16.074 


20 


1322.53 


7-56i 


3-78i 


30 


603 . 80 


16.562 


8 


281 


40 


3°S • 30 


32-436 


16.21S 


25 


1297.58 


7.707 


3-853 


35 


598.57 


16.707 


8 


353 


50 


3°5 • 60 


32.723 


16.361 


3° 


1273-57 


7.852 


3.926 


40 


593-42 


16.852 


8 


426 










35 


1250.42 


7-997 


3-999 


45 


588.36 


16.996 


8 


498 


19 


302.94 


33.010 


16.505 


40 


1228. 11 


8.143 


4.071 


50 


583.38 


17. 141 


8 


57i 


IO 


300.33 


;;.2o" 


16.64S 


45 


1206.57 


8.288 


4.144 


55 


578.49 


17.286 


8 


643 


20 


297.77 


33-5^; 


16.792 


50 


1185.78 


8-433 


4.217 












30 


295.25 


33-870 


16.93s 


55 


1165.70 


8-579 


4.289 


10 


573-69 


I7-43I 


8 


716 


40 


292.77 


34-157 


17.078 










10 


564-3 1 


17.721 


8 


860 


50 


290.33 


84-443 


1 -.222 


5 


1146.28 


8.724 


4.362 


20 


555-23 


18. on 


9 


005 










5 


1127.50 


8.869 


4-435 


30 


546.44 


18.300 


9 


150 


20 


287.94 


34.730 


17.363 


10 


1109.33 


9.014 


4-507 


40 


537-92 


18.590 


9 • 295 











<34 



TABLES AND FORMULAS. 



MOMENTS OF INERTIA. 




bd 



d* 



d*-df 



bd-b x d x 



■bd 



td+t,b 



bd-b,d. 



■bd 3 



12 



12 



a? < 



rorrf 



■(d*-d i *) 



^ibd'-b^dS) 



-bd* 



l^tdi+btS) 



Kbd* - ^flV> 2 -4ddd l d 1 (d-d 1 ) i \d_ , ^</,/ </- < /i \ 
12(bd-bidJ 2 a ^bd~b l d l f 



9. Circle 



10. Hollow 
Circle.. 



11. Ellipse 



12. Hollow 
Ellipse 



f* 


64 


^(</ 2 - ^! 2 ) 
4 


iridt-d!*) 

64 


f" 


7r£rf 3 

64 


£.0*-^) 


TrC^ 3 -^! 3 ) 
64 



i* 



TABLES AND 
BENDING MOMENTS 



FORMULAS. 65 

AND DEFLECTIONS. 



Manner of Supporting 
Beams. 



h- — i 



® 



pDQQQQQQGQ 




—l—. 



o cccax xyx ^B , 



Maximum 

Bending 

Moment, M. 



Wl 



W.l^+W^ 



wl* 



7Vl 2 



+ WI 



Wl 
4 



IV 



/,/. 



wl* 

~8~ 



-Wl 



wl 2 



Wl 
8 



wl* 

12 



Maximum 
Deflection, s. 



1 ?T/3 
3 i:V 



Remarks. 



8 EI 



1 J77 3 1 *T' I 3 
3 ^7 ' 8 ^7 



4& El' 



5 ?P/3 
384 TiV 



Wl 3 

.0182-^7- 

EI 



.(KIM 






193 z?y 



384 ijV 



Cantilever, load at free 
end. 



Cantilever, more than 
one load. 



Cantilever, uniform 
load ey lb. per unit 
of length, -fi 7 ' = zy /. 



Cantilever, load partly 
uniform, partly con- 
centrated. 



Simple beam, load at 
middle. 



Simple beam, load at 
some other point 
than the middle. 



Simple beam, uni 
formly loaded. 



One end fixed, other 
end supported, load 
in the middle. 



One end fixed, other 
end supported, uni- 
formly loaded. 



Both ends fixed, load 
in the middle. 



Both ends fixed, uni- 
formly Loaded. 



CG 



TABLES AND FORMULAS. 



SPECIFIC GRAVITIES AND WEIGHTS 
PER CUBIC FOOT. 



METALS. 



Substance. 



Weight per 
Cubic Foot 
in Pounds. 



Osmium 

Platinum 

Gold 

Mercury 

Lead (cast). . . 

Silver 

Copper (cast) . 

Brass 

AVrought Iron 
Cast Iron 

Steel 

Tin (cast) 

Zinc (cast) . . . 
Antimony. . . . 
Aluminum . . . 



i<437-5 
!,343- 8 
1,218.8 

850.0 
709.4 
656.3 
549-4 

523-Z 
480.0 
450.0 
490.0 

455- 6 
428.8 
419.4 
i5 6 -3 



WOODS. 



Substance. 



Weight' per 
Cubic Foot 
in Pounds. 




Ash 

Beech 

Cedar 

Cork 

Ebony (American) 

Lignum-vitae 

Maple 

Oak (old) 

Spruce 

Pine (yellow) 

Pine (white) 

Walnut 



TABLES AND FORMULAS. 
LIQUIDS. 



07 



Substance. 



Specific 
Gravity. 



Weight 

per 

Cubic Foot 

in Pounds. 



Acetic Acid , 

Nitric Acid 

Sulphuric Acid 

Muriatic Acid 

Alcohol 

Turpentine 

Sea Water (ordinary) 
Milk 



1.062 
1. 217 
1. 841 
1. 200 
.800 
.870 
1.026 
1.032 



66.4 

76.1 

115. 1 

75-° 
50.0 

54-4 
64. 1 

64-5 



GASES. 

At 32° F., and under a Pressure of One Atmosphere. 



Substance. 



Specific 
Gravity. 



Weight 

per 

Cubic Foot 

in Pounds. 



Atmospheric Air 

Carbonic Acid 

Carbonic Oxide 

Chlorine 

Oxygen 

Nitrogen 

Smoke (bituminous coal) 

Smoke (wood) 

*Steam at 212° F 

Hydrogen 



1. 0000 
1.5290 
.9674 
2.4400 
1. 1056 

•973 6 
. 1020 
.0900 
.4700 
.0692 



08073 
12344 
07810 
19700 
0S925 
07860 
00815 
00727 
03790 
00559 



* The specific gravity of steam at any temperature and pressure com- 
pared with air at the same temperature and pressure is 0.62& 



G8 



TABLES AND FORMULAS. 
MISCELLANEOUS. 



Substance. 



Emery 

Glass (average) 

Chalk. 

Granite 

Marble 

Stone (common) 

Salt (common) 

Soil (common) 

Clay 

Brick 

Plaster Paris (average) 
Sand 




Weight 

per 

Cubic Foot 

in Pounds. 



250 

175 
174 
166 
169 
i 5 3 

133 
124 
121 
118 
I2 5 
Ir 3 



COEFFICIENTS FOR FLOW OF WATER 



DISCHARGE OF STANDARD ORIFICES. 



COEFFIC1E 


;xts 


FOR CIRCULAR VERTICAL ORIFICES. 


Head h 




Diameter 


of Orifice in Feet. 






in Feet. 
















c 


).02 


0.04 


0.07 


0. 10 c 


). 20 c 


3. 60 ] 


[. 00 


0.4 




0.637 


0.624 


0.618 








0.6 


655 


.630 


.618 


.613 


601 


593 




0.8 


648 


.626 


.615 


.610 


601 


594 


59o 


1.0 


644 


.623 


.612 


.608 


600 


595 


59i 


i-5 


637 


.618 


.608 


.605 


600 1 


59 6 


593 


2.0 


632 


.614 


.607 


. 604 


599 


597 


595 


2-5 


629 


.612 


.605 


.603 


599 


598 


59 6 


3-° 


627 


.611 


. 604 


.603 


599 


598 


597 


4.0 


623 


. 609 


.603 


. 602 


599 


597 


59 6 


6.0 


6l8 


. 607 


. 602 


. 600 


598 


597 


596 


8.0 


614 


.605 


.601 


. 600 


593 


59 6 


596 


10. 


6ll 


.603 


•599 


■593 


597 


59 6 


595 


20.0 


6oi 


•599 


•597 


•596 


59 6 


59 6 


594 


50.0 


59 6 


•595 


•594 


•594 


594 


594 


593 


100. 


593 


•59 2 


•592 


•H 


59 2 


59 2 


59 2 



TABLES AND FORMULAS. 



69 



COEFFICIENTS FOR SQUARE VERTICAL ORIFICES. 



Head h 


Side of the Square in Feet. 


in Feet. 


0.02 


0.04 


0.07 


0. 10 


0. 20 


0. 60 


1. 00 


0.4 
0.6 
0.8 
1.0 

i-5 

2.0 

2-5 

3-° 

4.0 

6.0 

8.0 

10. 

20.0 

50.0 

100. 


0.660 
.652 
.648 
.641 
•637 
•634 
.632 
.628 
.623 
.619 
.616 
.606 
.602 
•599 


0.643 
.62,6 
.631 
.628 
.622 
.619 
.617 
.616 
.614 
.612 
.610 
.608 
.604 
.601 
•598 


0.628 
.623 
.620 
.618 
.614 
.612 
. 610 
.609 
.608 
.607 
.606 
.605 
.602 
.601 
•598 


0.621 
.617 
.615 
.613 
.610 
.608 
.607 
.607 
.606 
.605 
.605 
.604 
.602 
. 600 
•598 


0.605 
.605 
.605 
.605 
.605 
.605 
.605 
.605 
.604 
.604 
.603 
.602 
.600 
•598 


0.598 
.600 
.601 
.602 
.604 
.604 
.604 
.603 
.603 
.603 
. 602 
. 601 

•599 
•598 


°-597 
•599 
.601 
.602 
.602 
.603 
.602 
.602 
.602 
.601 
.600 

•599 
•598 



COEFFICIENTS FOR RECTANGULAR ORIFICES 
1 FOOT WIDE. 



Head h 






Depth of Orifice 


in Feet 






on Center 
















of Orifice 
















in Feet. 


0.125 c 


).25 


0.50 


o-75 


1. 00 


1-5° 


2.00 


0.4 


0.634 


633 


0.622 










0.6 


■ 633 


633 


.619 


0.614 








0.8 


■ 633 


633 


.618 


.612 


0.608 






1.0 


.632 


632 


.618 


.612 


.606 


0.626 




!-5 


.630 


631 


.618 


.611 


.605 


.626 


0.628 


2.0 


.629 


630 


.617 


.611 


.605 


.624 


.630 


2-5 


.628 


628 


.616 


.611 


.605 


.616 


.627 


3-° 


.627 


627 


•615 


.610 


.605 


.614 


.619 


4.0 


.624 


624 


.614 


. 609 


.605 


.612 


.616 


6.0 


.615 


615 


.609 


.604 


.602 


.606 


.610 


8.0 


.609 


607 


.603 


.602 


.601 


.602 


.604 


10. 


.606 


603 


. 601 


. 60 1 


. 601 


.601 


.602 


20.0 








. 601 


.601 


.601 


.602 



vO 



TABLES AND FORMULAS. 



DISCHARGE OF WEIRS. 



COEFFICIENTS FOR WEIRS WITH END CONTRACTIONS. 



Effective 




Length of Weir 


in Feet. 






Head in 
















Feet. 


0.66 


1 


2 


3 


5 


10 


!9 


O. I 


0.632 


639 


646 


652 


0-653 


-655 


0.656 


0.15 


.619 


625 


634 


638 


.640 


.641 


.642 


0. 20 


.611 


618 


626 


630 


.631 


-633 


•634 


0.25 


.605 


612 


621 


624 


.626 


.628 


.629 


0.30 


.601 


608 


616 


619 


.621 


.624 


.625 


0.40 


•595 


601 


609 


613 


.615 


.618 


. 620 


0.50 


•59° 


596 


605 


608 


.611 


.615 


.617 


0.60 


.587 


593 


601 


605 


.608 


.613 


.615 


0. 70 




59° 


598 


603 


.606 


.612 


.614 


0.80 






595 


600 


. 604 


.611 


.613 


0.90 






592 


598 


.603 


.609 


.612 


1. 00 






59° 


595 


.601 


.608 


.611 


1.2 






5*5 


59i 


•597 


.605 


.610 


1.4 






580 


587 


•594 


.602 


.609 


1.6 








582 


•59 1 


.600 


.607 



Note. — The head given is the effective head, H + 1.4 h. When the 
velocity of approach is small, h is neglected. 



COEFFICIENTS FOR WEIRS WITHOUT END CONTRACTIONS. 


Effective 
Head in 






Length 


of Weir 


in Feet. 




















Feet. 


*9 


10 


7 


5 


4 


3 


2 


0. 10 


0.657 


0.658 


0.658 


0.659 








0.15 


•643 


.644 


•645 


•645 


0.647 


.649 


652 


0. 20 


• 6 35 


•637 


•637 


.638 


.641 


642 


645 


0.25 


.630 


.632 


^33 


•634 


.636] 


638 


641 


0.30 


.626 


.628 


.629 


.631 


■633 


636 


639 


0.40 


.621 


.623 


.625 


.628 


.630 


^33 


636 


0.50 


.619 


.621 


.624 


.627 


.630 


6 33 


637 


0.60 


.618 


. 620 


.623 


.627 


.630 


634 


63S 


0. 70 


.618 


. 620 


.624 


.628 


.631 


635 


640 


0.80 


.618 


. 621 


.625 


.629 


■633 


637 


643 


0.90 


.619 


.622 


.627 


.631 


•635 


639 


645 


1. 00 


.619 


. 624 


.628 


■^33 


•637 


641 


648 


1.2 


.620 


.626 


.632 


.636 


.641 


646 




1.4 


.622 


.629 


•634 


. 640 


.644 






1.6 


.623 


.631 


.637 


.642 


.647 







Note. — The head given is the effective head, H 
velocity of approach is small, h may be neglected, 



Jl When the 



TABLES AND FORMULAS. 





= 

= 

P 

« . 

P 

CD y 

y« 

J 

§ s 

°S 

fa 

in 

« 

faE 
fa& 

a - 

fa M 
fa W 
fa 

O 

fa 
H 
H 

fa 


fa 
P 
fa 



c 

o 
o 

'Jl 

u 

<u 

(X 

<D 

'o 
c 

> 


» 


inmo mo cor^o cor^O r— O co O 

Omcocinnoo Oco i^ r> in m -f tn 

COClddClClClMMIlUMMIIMM 

OOOOOOOOOOOOOOOO 


O O 

Cl Cl 

H 11 

o o 








N 


CO Cl COO H OO TtO OcOO COO oo 
OO^-cicimmo ooo oo r^o o -t tn 

CO Cl Cl Cl Cl Cl Cl Cl M M M H M M M M 

OOOOOOOOOOOOOOOO 


^o ^o ^ 

Cl Cl w 

OOO 











in r^ go O N in inco O cor^O *~^0 coO 
oo-3-cocicimoo ooo co o o ir> rr 

OOOOOOOOOOOOOOOO 


OOO 
co co ci 

OOO 


O 

o 




00 


vni--r^ci go OO OO cor^O t-^O coO 
m r^ in ~f co co ci m m O O O i^ r^ O in 

CO Cl Cl Cl Cl Cl Cl Cl Cl Cl H H H M M M 

OOOOOOOOOOOOOOOO 


O in in 

-f co ci 

OOO 


in 

O 




^o 


vnr^t^u-.o r-r^o O cor^o r^o coO 
Nooo in in -|- co co ci h o Ococo r^O 

CO Cl Cl Cl Cl Cl Cl Cl Cl CM Cl Cl M M M M 

OOOOOOOOOOOOOOOO 


ooo 

in rf co 

OOO 


OOO 

Cl Cl M 

OOO 




in 


O CO O O CO O ih lfivO O M 1^ to N O ""> 
<tCM>inir)inttcnN <M m O Oco co O 
CO Cl Cl Cl Cl M Cl Cl Cl Cl Cl Cl M w M M 

OOOOOOOOOOOOOOOO 


in in in in in in 
in ^ co ci ci m 

O O O O o o 




Th 


mO m to i^ to m O cor^o too to 1^ O O O O 
in w co O in in -t -t to ci ci m O Oco r^o in -r 

COCOCldClClClCIClClClClClMi-IMIiMM 

OOOOOOOOOOOOOOOOOOO 


ooo 

co co ci 

OOO 




ro 


<N in in in mco OO O cor^OI^-O to O O O O 
co co O co r-«o O ininrj-cocOM w O Oco O O 

CO CO CO Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl M M m n 

OOOOOOOOOOOOOOOOOOO 


OOO 
m t to 

OOO 




o 


in o in in <n co mco Ocor-^o^OdO 
toNtoo Oco r^oo in t -tM ci m o 
"3-cocococicicicicicicicicicicici 

OOOOOOOOOOOOOOOO 


o o o o o o 

Ooo r^o in Tf 

o o o o o o 




- 


inOOvoOOOinOin inco co O to O 
O "-j-OO inrtCN Oooo m -t to to ci h 
intt-tcototototoN ci ci ci ci ci ci ci 

OOOOOOOOOOOOOOOO 


ooo 

O Ooo 

Cl M W 

ooo 


OOO 
r^ O in 

OOO 






3 

(Li 

s 

s 


o 


O O co coO co O co co O 
Oinco coo co OcoOcoOO 
MCMcoininO co OcoOcoOOO 
-rOcoci'i-Oinco OcoOcoOOO 
OOOmumcmco ino co O co inO O 


o o 

ooo 

mom 


OOO 

c c 


H H H 11 Cl 

II II II II II II II II II II II II II II II II 

H M n CN CO rtO CO O Cl O CO O "*■ 
H H H M C^ Cl 


ci co co T ino 

II II II II II II 

O O ci oo O ci 
co co rf *3-0 r^ 1 

1 



M. 



7:1 



TABLES AND FORMULAS. 



COEFFICIENTS FOR ANGULAR BENDS. 

a = angle of be?id in degrees. 




COEFFICIENTS FOR CIRCULAR BENDS. 

r = radius of pipe. R = radius of bend. 



r 
R~ 


. i 


2 


•3 


•4 


•5 


.6 


•7 


.8 


•9 


1 .0 


c' = 


•I3 1 


.138 


.158 


. 206 


.294 


.440 


.661 


•977 


1 .408 


1.978 



COEFFICIENTS FOR DARCY'S FORMULA. 



Diameter 

of Pipe 

in Inches. 


Coefficients for 


Coefficients for 


Rough Pipes. 


Smooth Pipes. 


3 


0.00080 


0.00040 


4 


.00076 


.00038 


6 


.00072 


.00036 


8 


.00068 


.00034 


10 


.00066 


.00033 


12 


.00066 


.00033 


14 


.00065 


.00033 


16 


.00064 


.00032 


24 


.00064 


.00032 


3° 


.00063 


.00032 


3^ 


.00062 


.00031 


48 


.00062 


.00031 



TABLES AND FORMULAS. 



73 



THE PROPERTIES OF SATURATED 
STEAM. 



O vU 

rt £ 
>3 

?(? 

s ^ 

,o cu 

rt Oh 

m 5 
o O 
VHpLn 


+-> 

Em CU 
0) 

£ & 

u 
cu 

s 

cu 


Quantities of Heat in British 
Thermal Units. 


4-1 
O 

O 
O W 

3d, 

o| 

-m cu 

§03 

"cu 


Volume. 


cu cu • 

rt^ 
£!? 

ts s 

^ % u 

•M ^£ 


CU . 

+j cu 

1-4 Si 
_^ 

O 


M 

> 

O 
O. 
05 
+-> 
d 
cu 

w 



Eh 


B 

<u . 

CO a; 
cu 

O 


B a>, 

^ Q 
o^-^ 

O •—* . ^ 

•M O +Sg 

03 ^ V? 4-1 


I 


2 


3 


4 


5 


6 


7 


8 


/ 


* 


? 


Z 


H 


[F 


F 


7? 


I 

2 

3 

4 
5 

6 

7 
8 

9 

IO 

ii 

12 

J 3 
14 

14.69 

15 
16 

17 
18 

19 


102.018 
126.302 
141.654 
153.122 
162.370 

170.173 
176.945 
182.952 
188.357 
193.284 

197.814 
202.012 
205.929 
209.604 

212.000 

213.067 
216.347 
219.452 
222.424 

225.255 


70.040 

94.368 

109. 764 

121. 271 

130.563 

138.401 
I45-2I3 
r 5!-255 
156.699 
161.660 

166. 225 

170.457 
174.402 
178. 112 

180.531 

181.608 
184.919 

188.056 
191.058 
193.918 


1043.015 
1026.094 
1015.380 
1007.370 
1000.899 

995-441 
990.695 
986.485 
982.690 
979.232 

976.050 
973.098 
970.346 
967.757 

966.069 

965.318 
963.007 
960.818 
958.721 
956.725 


1113-055 
1120.462 
1125.144 
1128.641 
1131.462 

1133.842 
1135.908 
1137.740 

II39-389 
1 140.892 

1142.275 

^43-555 
1144.748 
1145.869 

1 146.600 

1146.926 
1147.926 
1148.874 
1149.779 
1150.643 


.003027 
.005818 
.008522 
.011172 
.013781 

•016357 
.018908 
.021436 
.023944 

•026437 

.02891 1 
.031376 
.033828 
•036265 

•037928 

.038688 
.041 109 

•°435 I 9 
.045920 
.048312 


33°-4 
I7I-9 
U7-3 
89.51 

72.56 

61. 14 
52.89 
46.65 
4i-77 
37-83 

34-59 
31.87 
29.56 

27-58 
26.37 

25.85 
24-33 
22.98 
21. 78 

20. 70 


20623 
10730 

7325 
5588 

453° 

3816 
3302 
2912 
2607 
2361 

2159 
1990 

1845 
1721 

1646 

1614 

*5 X 9 

1434 
1359 
1292 



TABLES AND FORMULAS. 



I 


2 


3 


4 


5 


6 


7 


8 


p 


t 


q 


L 


H 


W 


V 


R 


20 


227.964 


196.655 


954.814 


1151.469 


.050696 


I 9-73 


1231.0 


22 


233.069 


201.817 


951.209 


1153.026 


.055446 


18.04 


1126.0 


24 


237.803 


206.610 


947.861 


1154.471 


.060171 


16.62 


1038.0 


26 


242.225 


211.089 


944-73° 


1155.819 


.064870 


15.42 


962.3 


28 


246.376 


215.293 


941.791 


1157.084 


•069545 


14-38 


897.6 


30 


250.293 


219. 261 


939.019 


1158.280 


.074201 


13-48 


841.3 


3 2 


254.002 


223.021 


936.389 


1159.410 


.078839 


12.68 


791.8 


34 


257.5 2 3 


226.594 


933-89I 


1160.485 


.083461 


11.98 


748.0 


36 


260.883 


230.001 


931.508 


1161.509 


.088067 


11.36 


708.8 


38 


264.093 


233.261 


929.227 


1162.488 


•092657 


10.79 


673-7 


40 


267.168 


236.386 


927.040 


1163.426 


.097231 


10.28 


642.0 


42 


270. 122 


239-389 


924.940 


1164.329 


.101794 


9.826 


6i3-3 


44 


272.965 


242.275 


922.919 


1165.194 


.106345 


9-403 


587-o 


46 


275.704 


245.061 


920.968 


1166.029 


.110884 


9.018 


563-0 


48 


278.348 


247.752 


919.084 


1166.836 


.115411 


8.665 


54o.9 


5° 


280.904 


2 5°-355 


917. 260 


1167.615 


.119927 


8.338 


520.5 


5 2 


283.381 


252.875 


9*5-494 


1168.369 


•124433 


8.037 


501.7 


54 


285.781 


255-3 21 


913.781 


1 169. 102 


.128928 


7-756 


484.2 


56 


288.111 


257.695 


912. 118 


1169.813 


•I334I4 


7.496 


467.9 


58 


290.374 


260.002 


910.501 


1170.503 


.137892 


7.252 


452.7 


60 


29 2 -575 


262.248 


908.928 


1 171. 176 


.142362 


7.024 


438.5 


62 


294.717 


264.433 


907.396 


1171.829 


. 146824 


6. 811 


425.2 


64 


296.805 


266.566 


905.900 


1172.466 


.151277 


6.610 


412.6 


66 


298.842 


268.644 


904.443 


1173.087 


•I5572I 


6.422 


400.8 


68 


300 831 


270.674 


903.020 


1173.694 


.160157 


6.244 


389.8 


70 


302.774 


272.657 


901.629 


1174.286 


.164584 


6.076 


379-3 


72 


304.669 


274.597 


900.269 


1174.866 


. 169003 


5-9I7 


369-4 


74 


306.526 


276.493 


898.938 


H75-43 1 


•I734I7 


5-767 


360.0 


76 


3°8-344 


278.350 


897-635 


II75-985 


.177825 


5.624 


35 1 - 1 


78 


310.123 


280. 1 70 


896.359 


1176.529 


. 182229 


5.488 


342.6 


80 


311.866 


281.952 


895.108 


1177.060 


.186627 


5-358 


334-5 


82 


3J3-576 


283.701 


893.879 


1177.580 


. 19101 7 


5-235 


326.8 


84 


3 I 5- 2 5° 


285.414 


892.677 


1178.091 


.195401 


5.118 


3*9-5 


86 


316.893 


287.096 


891.496 


1178.592 


.199781 


5.006 


3 I2 -5 


88 


318.510 


288.750 


890-335 


1179.085 


.204155 


4.898 


305-8 



TABLES AND FORMULAS. 



75 



I 


2 


3 


4 


5 


6 


7 


8 


p 


/ 


(/ 


L 


// 


IV 


V 


R 


90 


320.094 


290.373 


889.196 


1179.569 


.208525 


4.796 


299.4 


92 


321.653 


291.970 


888.075 


1180.045 


. 212892 


4.697 


293.2 


94 


323- l8 3 


293-539 


886.972 


1180.511 


•217253 


4.603 


287.3 


96 


324.688 


295.083 


885.887 


1 180.970 


. 221604 


4.513 


281.7 


98 


326. 169 


296.601 


884.821 


1 181.422 


.22595c 


4.426 


276.3 


100 


327.625 


298.093 


883.773 


1181. 866 


.230293 


4-342 


271. 1 


105 


331.169 


301.731 


881.214 


1182.945 


.241139 


4.147 


258.9 


no 


334-582 


305.242 


878.744 


1183.986 


.251947 


3-969 


247.8 


ii5 


337-874 


308.621 


876.371 


1184.992 


. 262732 


3.806 


237.6 


120 


341.058 


311.885 


874.076 


1185.961 


•2735oo 


3-656 


228.3 


125 


344- 1 3 6 


3I5-05 1 


871.848 


1186.899 


•284243 


3-5i8 


219.6 


130 


347.121 


318. 121 


'869.688 


1187.809 


. 294961 


3-39° 


211. 6 


135 


35°- OI 5 


321.105 


867.590 


1188.695 


•305659 


3.272 


204.2 


140 


352.827 


324.003 


865.552 


1189.555 


■3 l6 33 s 


3. 161 


*97-3 


145 


355-562 


326.823 


863.567 


1190,390 


.326998 


3-058 


190.9 


150 


35 8 - 22 3 


329.566 


861.634 


1191.200 


•337643 


2.962 


184.9 


160 


3 6 3'346 


334.850 


857.912 


1 192. 762 


.358886 


2.786 


173-9 


170 


368.226 


339-892 


854.359 


1194-251 


.380071 


2.631 


164.3 


180 


372.886 


344.708 


850.963 


1195.671 


.401201 


2-493 


155-6 


190 


377-352 


349-329 


847.703 


1197.032 


.422280 


2.368 


147.8 


200 


381.636 


353-766 


844.573 


H98.339 


•4433 10 


2.256 


140.8 


210 


385-759 


358.041 


841.556 


JI 99-597 


.464295 


2.154 


J 34-5 


220 


389-736 


362.168 


838.642 


1200.810 


.485237 


2.061 


128.7 


230 


393-575 


366.152 


835.828 


1 201.980 


.506139 


1.976 


I2 3-3 


240 


397-285 


370.008 


^33- io 3 


1203. in 


•527003 


1.898 


118.5 


250 


400.883 


373-75° 


830.459 


1204.209 


•547831 


1.825 


114. 


260 


404.370 


377-377 


827.896 


1205.273 


.568626 


1-759 


109.8 


270 


407-755 


380.905 


825.401 


1206.306 


•589390 


1.697 


105.9 


280 


411.048 


384-337 


822.973 


1207.310 


.610124 


1.639 


102.3 


290 


414.250 


387.677 


820.609 


1208.286 


.630829 


1-585 


99.0 


300 


4I7-37I 


39°-933 


818.305 


1209.238 


.651506 


r -535 


95. S 



TABLES AND FORMULAS. 



MISCELLANEOUS TABLES. 



STANDARD DIMENSIONS OF WROUGHT-IRON 
STEAM, GAS, AND WATER PIPES. 



Nominal 
Diameter . 

T U m 

in Inches. 


ickness 
[nches. 


Actual 
Internal 
Diameter 
in Inches. 


Actual 
External 
Diameter 
in Inches. 


Threads 
per Inch. 

71 


Pitch of 
Threads. 


i 


068 


. 270 


•405 


27 


•037 


i 


088 


•3 6 4 


•540 


l8 


.056 


t 


091 


•494 


•675 


l8 


.056 


l 
2 


109 


.623 


.840 


14 


.071 


f 


TI 3 


.824 


1.050 


14 


.071 


I 


r 34 


1.048 


I-3I5 


Il£ 


.087 


ii 


140 


1.380 


1. 660 


"i 


.087 


ii 


i45 


1. 611 


1. 900 


ni 


.087 


2 


i54 


2.067 


2 -375 


"* 


.087 


*i 


204 


2.468 


2.875 


8 


< I2 5 


3 


217 


3.061 


3-5°° 


8 


.125 


-3* 


226 


3-54S 


4.000 


8 


•125 


4 


237 


4.026 


4.500 


8 


.125 


4* 


247 


4.508 


5.000 


8 


•125 


5 


2 59 


5-°45 


5-5 6 3 


8 


.125 


6 


280 


6.065 


6.625 


8 


• I2 5 


7 


301 


7.023 


7.625 


8 


•125 


8 


322 


. 7-932 


8.625 


8 


• I2 5 


9 


344 


9.001 


9.688 


8 


.125 


TO 


3 66 


10.019 


10.750 


8 


•125 



TABLES AND FORMULAS. 



STANDARD PIPE FLANGES. 



Inside 
Diam. 

of 
Pipe. 


Thick- 
ness of 


Diam. 
of 


Length 
of 


No. of 
Bolts. 


Thick- 
ness of 


Diam. of 
Bolt 


Diam. 
of 


Pipe. 


Bolts. 


Bolts. 


Flange. 


Circle. 


Flange. 


2.0 


.409 


5 

8 


2.0 


4 


-f 


4-75 


6.0 


2-5 




429 


5 

8 


2.25 


4 


1 1 

1 6 


5-25 


7.0 


3-o 




448 


t 


2-5 


4 


f 


6.0 


7-5 


3-5 




466 


f 


2.5 


4 


1 3 
1 6 


6-5 


8-5 


4.0 




486 


1 


2.75 


4 


1 5 

16 


7-25 


9.0 


4 5 




498 


3 

4 


3-° 


8 


1 5 
T6 


7-75 


9-25 


5 




525 


1 


3-° 


8 


1 5 
T6 


8-5 


10. 


6 




563 


t 


3° 


8 




9.625 


11. 


7 




600 


I 


3- 2 5 


8 


T 1 
I T6" 


io-75 


I2 -5 


8 




639 


I 


3-5 


8 


>i 


u-75 


*3-5 


9 




678 


1 


3-5 


1 2 


ii 


13.0 


r 5-° 


10 




713 


i 


3- 6 25 


12 


t 3 
M6 


14.25 


16.0 


12 




.790 


i 


3-75 


12 


T 1 

T 4 


16.5 


19.0 


14 




864 




4-25 


12 


if 


18.75 


21.0 


15 




904 




4-25 


16 


if 


20.0 


22.25 


16 




.946 




4-25 


16 


T 7 
I T6 


21.25 


23-5 


18 


1 


020 


1 8 


4-75 


16 


t 9 
1 1 6 


22.75 


25.0 


20 


1 


.090 


"i 


5"° 


20 


t 1 1 
1 1 6 


25.0 


27-5 


22 


1 


.180 


T 1 
I 4 


5-5 


20 


t 1 3 
1 1 6 


27.25 


29-5 


24 


1 


.250 


T 1 
x 4 


5-5 


20 


1* 


29-5 , 


32.0 


26 


1 


300 


T l 
1 ± 


5-75 


24 


2 


3i-75 


34- 25 


28 


1 


.380 


T l 


6.0 


28 


' *A 


34-o 


3 6 -5 


3° 


1 


.480 


'I 


6.25 


28 


H 


36.0 


38.75 


36 


1 


. 710 


T i 

i 8 


6-5 


32 


^4 

8 


42.75 


45-75 


42 


1 


870 


4 


7.25 


36 


2 | 


49-5 ' 


52-75 


48 


2 


170 


T 1 
1 2 


7-75 


44 


2 4 


56.0 


59 5 



78 



TABLES AND FORMULAS. 



SPECIFIC HEAT OF SUBSTANCES. 



Substance. 


Specific 
Heat. 


Substance, 


Specific 
Heat. 


Water . 


1.0000 
. 2026 


Ice 


cmr, 


Sulphur 


Steam (superheated) 
Air 


3 


4805 

-375 

2175 
4090 
2479 
2170 
2438 


Iron 


.1138 
.0951 

.0570 
.0562 

-°333 
.0314 


Copper 


Oxygen 

Hydrogen 


Silver 


Tin 


Carbon monoxide . . 
Carbon dioxide .... 
Nitrogen 


Mercury 


Lead 









CONSTANTS FOR APPARENT CUT-OFFS USED 
IN DETERMINING M. E. P. 



Cut-off. 


Constant. 


Cut-off. 


Constant. 


Cut-off. 


Constant. 


# 


.566 


3 A 


.771 


% 


.917 


Vs 


.603 


• 4 


.789 


•7 


.926 


% 


•659 


% 


•847 


X 


•937 


•3 


.708 


.6 


•895 


.8 


•944 


/3 


•743 


~/8 


.904 


n 


•95 1 



RIVETED JOINTS OF BOILERS. 



Thick- 


Diam- 


Diam- 


Pitch. 


Efficiency 


of Joint. 


ness of 


eter of 
Rivet. 


eter of 
Hole. 








Plate. 










/ 


d 


Single. 


Double. 


Single. 


Double. 


% !l 


5 A" 


11" 

16 


2" 


11 



.66 


•77 


a 

1 


1 1" 
1 e" 


H" 


Z \ G 


3/8" 


.64 


.76 


y 


H" 


1 3" 
16 


2}i" 


z%" 


.62 


•75 


7 " 

1 G 


1 3" 
"1 G 


<-/ // 
/8 


2fV 


sH' 


.60 


• / 4 


%" 


H' 


1 5 " 
16 


*X' 


i%" 


•58 


•73 



TABLES AND FORMULAS. 79 

POSITIONS OF ECCENTRIC RELATIVE TO CRANK. 





Kind of 


Kind of 
Rocker- 


Angle Between 
Crank and 


Position of 
Eccentric Rela- 




Valve. 


Arm. 


Eccentric. 




tive to Crank. 


I.... 


Direct. . . 


Direct 


90 -(- angle 
advance. . 


of 


Ahead of crank. 


II... 


Direct. . . 


Reversing.. 


90 — angle 
advance. . 


of 


Behind crank. 


III. . 


Indirect . 


Direct 


90 — angle 
advance. . 


of 


Behind crank. 


IV... 


Indirect. 


Reversing.. 


90 -|- angle 
advance . . 


of 


Ahead of crank. 



DIAMETERS OF STEAM AND EXHAUST PIPES. 



Diam. of cylinder. . . 


10 


12 


14 


16 


18 


20 


22 


24 


26 


28 


3° 


Diam. of steam pipe . 


3 


3^ 


4 


A% 


5 


6 


6 


7 


7 


8 


9 


Diam. of exhaust pipe 


3/ 2 


4 


5 


6 


6 


7 


8 


9 


9 


9 


10 



PISTON SPEEDS OF STEAM ENGINES. 

Ft. per min. 

Small stationary engines 300 to 600. 

Large stationary engines 600 to 1,000. 

Corliss engines 400 to 750. 

Locomotives 600 to 1,200. 

RATIO OF GRATE AREA OF BOILER TO 
HORSEPOWER. 



Ratio. Average. 

Plain cylindrical 5 to . 7 

Flue [ to .5 

Multitubular \ to .6 

Water tube 3 

Vertical 6 to . 7 

Locomotive 01 to .06 



.6 

•45 

•5 

-> 
• j 

.65 



80 TABLES AXD FORMULAS. 



RATIO OF HEATING SURFACE TO GRATE AREA. 

Plain cylindrical 12 to 15 

Flue 20 to 25 

Multitubular 25 to 35 

Vertical 25 to 30 

Water tube 35 to 40 

Locomotive 50 to 100 

RATIO OF HEATING SURFACE TO HORSEPOWER. 

Plain cylindrical 6 to 10 

Flue 8 to 12 

Multitubular 14 to 18 

Vertical 15 to 20 

Water tube 10 to 12 

Locomotive 1 to 2 



TABLES AND FORMULAS. 



81 



z 

< 
z 

h 

o 
/> 

c 

e 





o 
o 

CI 




H 


CO 


04 


04 


r\ 


M 


w 


H 


O 


O 


On 


ON 


ON 


m 


en 


r». 


r-^ 


r-N 


r^ 






-1- 


'O 


CI 




O 


ON CO 


r^ 


v'J 


in 


CO 


04 




C) 


~ 


on 


r^ 


N.O 


in 






C( 


04 


04 


04 


CI 




















O 


q 


O 


o 


q 








ON 


M 


o 


O 


R 


ON 


ON 


On CO 


CO 


r-> 


r^ 


1^ 


NO 


NO 


in 


in 


in 


in 








CO 


CO 


04 


M 


on 


r^ 


NO 




Tf 


co 


04 


M 


'■■ 


ON 


ro 


r^ 


O 


in 








04 


04 


04 


04 


04 




















o 


q 


q 


q 


q 








r^ 


ON 


on 


on 


on 


r^ 


r^ 


r-^ 


NO 


NO 


in 


in 


in 


'St 


^t- 


co 


co 


CO 


CO 




u 




co 


04 




O 


ON 


CO 


r^ 


o 


in 


-t- 


co 


04 




() 


(-^ 


CO 


i> 


NO 






nO 




04 


04 


04 


04 






















u 


q 


q 


q 


q 








"St 


O 


in 


in 


in 


rf 


■*t 


-i- 


co 


CO 


n 


04 


04 


H 


M 


o 


O 


o 


o 








CO 


04 




O 


cy> 






o 


in 


•st 


CO 


04 




C5 


<> 


en 




',., 


in 








04 


04 


04 


04 






^ 




S 




2 


2 


^ 




o 


q 


q 


q 


q 




o 




M 


CO 


04 


04 


01 


M 


H 


M 


O 


o 


o 


ON 


ON 


co 


m 


r^ 


r^ 


r^ 


r^ 






CO 


04 




O 


rr> 


on 


r-» 


NO 


in 


-t 


C4 


M 


o 


ON 


CO 


r-^ 


NO 


in 


-t 




H 




04 


04 


04 


04 




















<) 


<> 




o 


O 






8 




r^ 


on 


on 


on 


on 


r^ 


r^ 


r^ 


NO 


o 


in 


in 


in 


"Sfr 


•st 


co 


CO 


CO 


co 






04 




o 


~- 


CO 


i- 


o 


in 


-r 


CO 


C4 




C> 


C7n 


on 


r^ 


Nf3 


in 


"»t 






04 


CI 


CI 






















O 


O 


q 


U 


q 


q 








-t 


O 


m 


in 


in 


•st 


■*t 


-?t 


CO 


co 


CI 


04 


01 


M 


M 


o 


o 


o 


o 




o 




Cl 




O 


on 


CO 




o 


in 


~r 


co 


C4 




) 


r* 


a) 


i- 


NO 


m 


Tf 






CM 


04 


04 






















C) 


O 


o 


o 


O 




<u 






01 


Tt 


CO 


co 


co 


0) 


n 


CI 


M 


H 


o 


o 


o 


ON 


C?N 


oo 


CO 


on 


co 




o 

CO 




04 




o 


ON 


co 


j-^ 


NO 


U) 


-t 


CO 


04 




o 


or) 


r-> 


NO 


in 


-T 


CO 


CO 


a 

> 


04 


04 


CI 






















O 


O 


o 


O 


o 


O 


u 












































— 


w 








































Ph 










































<+4 


O 


M 


O 


o 


O 


ON 


ON 


ON 


cn 


CO 


1-* 


r^ 


r~^ 


NO 


NO 


in 


in 


in 


in 


0) 


O 








<) 


o> 


CO 


vO 


If) 


-st 


CI 


U 




O 


On CO 


r-> 


NC 




-r 


co 




CO 


04 


04 


CI 


*~l 


^ 


■"J 


* 


M 


*~l 


M m 


1 


1 


O 


O 


c> 


C) 


O 


() 


O 




— 


o 




















































































r^ 


On 


en 


on 


on 


r^ 


r^ 


r-i 


NO 


NO 


in 


in 


in 


■st 


•<st 


co 


co 


CO 


CO 




O 

-o 




o 


On 


on 


r^ 


NO 


in 


<St 


co 


04 




O 


C7n 


on 


r> 


vO 


in 


-t- 


CO 




04 


04 






















O 


O 


O 


q 


q 


q 


q 








"tf- 


vO 


in 


in 


in 


'st 


-1- 


^t 


co 


CO 


04 


04 


04 


M 


M 


o 


o 


o 


o 




o 

IT, 








O 


co 


r->. 


NO 


m 


"* 


CO 


01 




o 


CJN 


on 


r^. 


n£J 




Tt 


co 






04 


04 


M 










H 




M 


M 


M 




O 




















-t" 


CO 


co 


CO 


04 


04 


04 


M 


M 


O 


o 


o 


ON 


ON 


on 


on 


CO 


co 




IT) 

-T 






o 


ON 


co 


t^. 


NO 


in 


-st 


CO 


04 




<) 


C3N 


i- 


<c 


in 


"t 


co 


04 






C4 


04 




M 






H 




H 








O 


o 


o 


q 


q 


q 


q 








w 


rO 


04 


04 


04 


M 


M 


H 


o 


o 


ON 


ON 


ON 


co 


CO 


r- 


r^ 


t^ 


r^ 




o 
■"St 






o 


On 


CO 


1- 


V<) 


m 


-I- 


CO 


04 




ON 


CO 


i-^ 


o 


in 


-r 










0) 


04 




2 


2 


^ 


- 




M 










o 


















o 


H 


O 


O 


O 


On 


On 


ON 


CO 


CO 


r-^ 


r^ 


r^ 


NO 


NO 


in 


in 


\n 


in 








o 


O 


ON 


co 


I-* 


U-) 


^t 


co 


04 




n 


ON 


CO 


r^ 


<> 


in 


"St 


ro 


04 




co 




01 


CI 




M 
















o 


O 


O 


u 


q 


q 


o 


q 








vO 


on 


r^ 


r^ 


r^ 


NO 


nO 


NO 


in 


in 


<cf 


-St 


Tf 


CO 


CO 


04 


04 


04 


04 




o 

CO 




O 


o 


CO 


r^ 


o 




-t 


co 


04 


M 


O 


ON 


on 


r^ 




in 


•"St 


co 


04 






CI 






















o 


O 


o 


o 


q 


o 


q 


q 








-t 


o 


in 


in 


in 


Tt" 


'st 


-N+ 


CO 


co 


04 


04 


04 


M 


H 


n 


n 


n 


o 








O 


on 


on 


r^ 


no 


in 


-r 


co 


04 




O 


<■> 


CO 






in 


-t 


CO 


04 




04 




04 






















o 


O 


o 


q 


9 




O 




1) 


0>C 


S 










































J 




r\ 


o 


O 


O 


o 


O 


o 


O 












o 














Spc 


£ 


CO 


o- 


in 


o 


1- 


CO 




O 












NO 








CJ 


04 


H 













































82 



TABLES AND FORMULAS. 



•S9ipuj 






































III J9}9 


CO 


HH 


Tf" 


1>~ 


O 


oo 


NO 


CN 


CM 


OO 


^ 


o 


NO 


CM 


00 


^t 


o 


NO 


I— i 


CM 


01 


CM 


OO 


OO 


oo 


oo 


"fr 


"*■ 


LO 


NO 


NO 


r- 


r^- 


00 


ON 


On 


-UIBIQ 




































•S9qDU£ 






















1 














'9JBnbg 


NO 


c« 


(N 


"fr 


l>- 


O 


CM 


LO 


00 


CO 


00 i Tt- 


On 


«fr 


O 


m 


o 


NO 


M 


M 


CM 


CM 


CM 


OO 


OO 


OO 


oo 


Th 


^-, u-> 


LO 


NO 


!>. 


t^- 


ZC 


00 


J9 a P!S 




















1 














'Id ' b S 


i~^ 


M 


<+ 


00 


M 


"fr 


l^ 


o 


O) t-^ 


o 


^ 


NO 


r-~ 


OO 


00 


00 


r^ 


UI B9jy 

jBtrpy 


f-^ 


^t 


I" 1 


Os 


on 


Cn 


o 


OO 


no 


LO 


On 


NO 


r-^ 


CM 


M 


^t 


t-< 


CM 


H 


CM* 


oo 


oo 


4 


LO 


t^ 


oo 


On 


01 


LO 


ON 


CO 


00 


CO 


co 


4 


6 




















1-1 


M 


* 


01 


CM 


CO 


CO 


^d- 


LO 


'Id ' D S 


i>» 


r^ 


oo 


00 


oo 


t^ 


t-« 


t^- 


NO 


"* 


H 


oo 


CO 


OO 


CO 


NO 


On 


H 


ui B9ay 

9AP99U3 


On 


"*d- 


o 


t^ 


LO 


Tj- 


-* 


LO 


!>• 


^h 


LO 


ON 


OO 


O 


!>. 


t^ 


1-1 


o 


6 


i_3 


CM* 


CM 


CO 


4 


LO 


NO 


I>- 


6 


CO 


NO 


6 


LO 


ON 


4- 


6 


NO 






















M 




CM 


CM 


01 


oo 


^ 


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TABLES AND FORMULAS. S3 

RULES AND FORMULAS. 



FORMULAS USED IN ALGEBRA. 

Let a and b be any two quantities, then, 

(a + b) 2 =- a' + 2ab + b\ (1.) Art. 432. 

{a - b) 2 = a 2 - 2ab + b\ (2.) Art. 432. 

(a + b)(a - b) = a" - b\ (3.) Art. 432. 

a*+2ab+b 2 = ( a +b)(a+b) = (a+b)\ (4.) Art. 455. 

cf-2ab+b* = (a-b)(a-b) = (a-b)\ (5.) Ait. 455. 

a *-b* = (a + b)(a-b). (6.) Art. 462. 

Let ax 2 -\- bx = e be any quadratic equation (it must be 
borne in mind that b and c may be positive or negative) ; 

then, 



—£±m+i== ±A € ± ^- «■««. 



THE TRIGONOMETRIC FUNCTIONS. 

Art. 98. 

~ . side opposite . , 

Sine — —. — ; therefore, 

hypotenuse 

Rule 1 . — Side opposite = hypotenuse x sine. 

_^ TT side opposite 

Rule 2.— Hypotenuse — ^A 1 

* sine 

~ . side adjacent , 

Cosine = — j f— / therefore. 

hypotenuse 



84 TABLES AND FORMULAS. 

Rule 3. — Side adjacent = hypotenuse X cosine. 

n . . jj . . side adjacent 

Rule 4. — Hypotenuse = 4 . 

cosine 

~ side opposite 

langent = -^ — t-f. - • therefore, 

side adjacent J 



Rule 5. — Side opposite = side adjacent x tangent. 

~ . side adjacent 

Cotangent — — t-. ± ,— • therefore, 

side opposite 

Rule 6. — Side adjacent = side opposite X cotangent. 



RULES FOR USING TABLE OF LOGARITHMS 
OF NUMBERS. 
Arts. 625-636. 

I. To find the Characteristic. — For a number greater 
than 1 the characteristic is one less than the number of in- 
tegral places in the number. For a number wholly decimal 
the characteristic is negative, and is numerically one greater 
than the number of ciphers between the decimal point and the 
first digit of the decimal. 

II. To find the Logarithm of a Number not hav- 
ing more than four figures. — Find the first three sig- 
nificant figures of the number zuhose logarithm is desired in 
the left-hand column ; find the fourth figure in the column at 
the top (or bottom) of the page, and in the column under (or 
above) this figure, and opposite the first three figures previously 
found, will be the mantissa, or decimal part, of the logarithm. 

TJie characteristic being found as described above, write it at 
the left of the mantissa, and the resulting expression will be 
the logarithm of the required number. 

III. To find the Logarithm of a Number con- 
sisting of five or more figures. 

(a) If the number consists of more than five figures, and 
the sixth figure is 5 or greater, increase the fifth figure by 1, 
and write ciphers in place of the sixth and remaining figures. 



TABLES AND FORMULAS. 85 

(b) Find the mantissa corresponding to the logarithm of 
the first four figures, and subtract this mantissa from the 
next greater mantissa in the table ; the remainder is the 
difference. 

(c) Find in the secondary table Jieaded P. P. a column 
headed by the same number as that just found for the differ- 
ence, and in this column opposite the number corresponding to 
the fifth figure (or fifth figure increased by 1) of the given 
number (this figure is always situated at the left of the 
dividing line of the column) will be found the P. P. (propor- 
tional part) for that number. The P. P. thus found is to be 
added to the mantissa found in (b), and tJie result is the 
mantissa of the logaritJim of the given number, as nearly as 
may be found with five-place tables. 

IV. To find a Number whose Logarithm is 
given. — 

(a) Consider the mantissa first. Glance along the 
different columns of the table which are headed O until the 
first two figures of the mantissa are found. Then glance 
down the same column until the third figure is found (or 1 
less than tJie third figure). Having found the first tJiree 
figures, glance to the right along the row in which they are 
situated until the last three figures of the mantissa are found. 
Then the number which lie ads the column in which the last 
three figures of the mantissa are found is the fourth figure 
of the required number, and the first three figures lie in the 
column Jieaded N, and in the same row in which lie the last 
three figures of tJie mantissa. 

(b) If the mantissa cannot be found in the table, find the 
mantissa which is nearest to, but less than, the given mantissa, 
and which call the next less mantissa. Subtract the next less 
mantissa from the next greater mantissa in the tabic to obtain 
the difference. Also subtract the next less mantissa from the 
mantissa of the given logaritJim, and call the remainder the 
P. P. Looking in the secondary table headed P. P. for the 
column Jieaded by the difference just found, find the number 
opposite tlie P. P. just found (or the P. P. corresponding most 



86 TABLES AND FORMULAS. 

nearly to that just found} ; this number is the fifth figure of 
the required number ; the fourth figure will be found at the 

top of the column containing the next less mantissa, and the 
first three figures in the column headed .V, and in the same 
row which contains the next less mantissa. 

(c) Having found the figures of the number as above 
directed, locate the decimal point by the rules for the charac- 
teristic, annexing cipliers to bring the number up to the re- 
quired number of figures if the characteristic is greater than Jf. 



RULES FOR USING TRIGONOMETRIC TABLES. 

Given, an angle, to find its sine, cosine, tangent, 
and cotangent. 

Rule 7. — Find in the table the sine, cosine, tangent, or co- 
tangent corresponding to the degrees and minutes of the angle. 

For the seconds, find the difference of the values of the sine, 
cosine, tangent, cr cotangent taken from the table, between 
which the seconds of the angle fall ; multiply this difference by 
a fraction whose numerator is the number of seconds in the 
given angle, and whose denominator is GO. 

If sine or tangent, add this correction to the value first found; 
if cosine or cotangent, subtract the correction. Art. 102 8 

Given, the sine, cosine, tangent, or cotangent to 
find the angle corresponding. 

To find the angle corresponding to a given sine, cosine, 
tangent, or cotangent whose exact value is not contained in 
the table: 

Rule 8. — Find the difference of the two numbers in the 
tabic between which the given sine, cosine, tangent, er co- 
tangent falls, and use the number of parts in this difference 
as the denominator of a fraction. 

Find the difference between the number belonging to the 
smaller angle, and the given sine, cosine, tangent, or cotangent, 
and use the number of parts in the difference just found as the 
numerator of the fraction mentioned above. Multiply this 
fraction by GO, and the result will be the number of seconds to 
be added to the smaller angle. Art. 1 05, 



TABLES AND FORMULAS. 87 
RULES FOR MENSURATION. 



THE TRIANGLE. 

Rule. — The area of any triangle equals one-half the 
product of the base and the altitude. Art. 118. 

Let a, b, and c be the lengths of the sides, s half the sum 
of these lengths, and A the area of any plane triangle; then, 

A — \/s (s — a) (s — b) (s — c), where s = . 

Art. 119. 



THE QUADRILATERAL. 

Rule. — The area of any parallelogram equals the product 
of the base and the altitude. Art. 1 29. 

Rule. — The area of a trapezoid equals one-half the sum of 
the parallel sides multiplied by t 'he altitude. Art. 130. 



THE CIRCLE. 

Rule. — The circumference of a circle equals the diameter 
multiplied by S. lj.16. Art. 131. 

Rule. — The diameter of a circle equals the circumference 
divided by S.I4I6. Art. 131. 

Rule. — The length of an arc of a circle equals the circum- 
ference of the circle of which the <xrc is a part multiplied by the 
number of degrees in the arc, and divided by 360. Art. 1 32. 

When only the chord of the arc and the height of segment 
are given, the following approximate formula may be used: 
Let c be the length of chord, h the height of segment, and / 
the length of arc; then, 



l= w*+i*-c m Anl33 

o 

To find the area of a circle: 

Rule. — Square the diameter, and multiply by .7834., or. 
square the radius and multiply by 8.I4.I6. Art, I 34. 

G. (1. IV— 28 



88 TABLES AND FORMULAS. 

Given, the area of a circle to find its diameter: 

Rule. — Divide the area by .785 J/., and extract the square 
root of the quotient. Art. 135. 

To find the area of a sector: 

Rule. — Divide the number of degrees in the arc of a sector 
by 360. Multiply the result by the area of the circle of which 
the sector is a part. Art. 137, 

Rule. — The area of a sector is equal to one-half the prod- 
uct of the radius and lengtli of arc. Art. 138. 

To find the area of a segment of a circle: 

Rule. — Draw radii from the center of the circle to the 
extremities of the arc of the segment; -find the area of the 
sector thus formed, subtract from this the area of the triangle 
formed by the radii and the chord of the arc of the segment, 
and the result is the area of the segment. Art. 139. 



THE ELLIPSE. 



To find the periphery (perimeter) of an ellipse: 

Let ti = 3.1416; C — periphery (perimeter) ; a = half the 

7 i i c 1 • • 7-. a — b 

major axis; o = halt the minor axis; D = 



a + b' 



rru r i , i\ 64 - 3 £> 4 

Then, C — tt (a -f- b) — 



64- WD 2 ' 
To find the area of an ellipse: 

Rule. — The area of an ellipse is equal to the product of its 
two diameters multiplied by .785J/.. Art. 144. 



THE PRISM AND CYLINDER. 

To find the area of the convex surface of any right prism 
or right cylinder: 

Rule. — Multiply the perimeter of the base by the altitude. 
Art. 158. 

To find the volume of a right prism or cylinder: 

Rule. — The volume of any right prism or cylinder equals 
the area of the base multiplied by the altitude. Art. 159. 



TABLES AND FORMULAS. 89 

THE PYRAMID AND CONE. 

Rule.— The convex area of a rigJit pyramid or cone equals 
the perimeter of the base multiplied by one-Jialf the slant 
height. Art. 164. 

Rule. — The volume of a right pyramid or cone equals the 
area of the base multiplied by one-third of the altitude. 
Art. 165. 

THE FRUSTUM OF A PYRAMID OR CONE. 

To find the convex area of a frustum of a right pyramid 
or right cone : 

Rule. — The convex area of a frustum of a right pyramid 
or right cone equals one- ha If the sum of the perimeters of 
its bases multiplied by the slant height of the frustum. 
Art. 169. 

To find the volume of the frustum of a pyramid or cone: 

Rule. — Add the areas of the upper base, the lower base, 
and the square root of the product of the areas of the two bases; 
multiply this sum by one-third of the altitude. Art. 170* 



THE SPHERE. 

Rule. — The area of the surface of a sphere equals the 
square of the diameter multiplied by 3.14-16. Art. 172. 

Rule. — The volume of a sphere equals the cube of the 
diameter multiplied by .5236. Art. 173. 

To find the diameter of a sphere of known volume: 

Rule. — Divide the volume by .5236 and extract the cube 
root of the quotient. Art. 174. 



FORMULAS USED I1V ELEMENTARY 
MECHANICS. 



UNIFORM MOTION. 

Let S = the length of space passed over uniformly; 

/ — the time occupied in passing over the space S; 
V = the velocity. 



90 TABLES AND FORMULAS. 

V=Y- ( 7 -) Art - 859 - 
5 = Vt. (8.) Art. 859. 

t = y. (9.) Art. 859. 



MASS, WEIGHT, AND GRAVITY. 

If the mass of the body be represented by 7/z, its weight 
by W, and the force of gravity at the place where the body 
was weighed by g t we have 

weight of body W /-***, v ^^^ 

mass = 2 — - — 2— -tt-j or m = ■ — . (lO.) Art. 888. 

force of gravity g ' 



FORMULAS FOR GRAVITY PROBLEMS. 

Let IV = weight of body at the surface; 

w = weight of a body at a given distance above or 

below the surface ; 
d — distance between the center of the earth and the 

center of the body; 
R = radius of the earth = 4,000 miles. 

Formula for weight when the body is below the surface: 

w R = d IV. (11.) Art. 891 . 

Formula for weight wmen the body is above the surface: 

wd*=WR\ (12.) Art. 891. 



FALLING BODIES. 

Let g= force of gravity = constant accelerating force due 
to the attraction of the earth ; 
/ = number of seconds the body falls; 
v = velocity at the end of the time t; 
Ji = distance that a body falls during the time t. 

v = gt. (13.) Art. 896. 

That is, the velocity acquired by a freely falling body at the 
end of t seconds equals 32. 16 multiplied by the time in seconds. 

f = *. (14.) Art. 896- 



TABLES AND FORMULAS. 01 

That is, the number of seconds during which a body must 
have fallen to acquire a given velocity equals the given velocity 
in feet per second divided by 32. 16. 

h = ^-. (15.) Art. 896. 

2g 

That is, the height from which a body must fall to acquire 
a given velocity equals the square of the given velocity divided 
by 2 X 32. 16. 



v = \l%gk. (16.) Art. 896. 

That is, the velocity that a body zvill acquire in falling 
through a given height equals the square root of the product 
of twice 32. 16 and the given height. 

h = \g?. (17.) Art. 896. 

That is, the distance a body will fall in a given time equdls 
32.16 -f- 2 multiplied by the square of the number of seconds. 

t -J%h; ( 18#) Art. 896. 

g 

That is, the time it will take a body to fall through a given 
height equals the square root of twice the height divided by 
32.16. 



CENTRIFUGAL FORCE. 

The value of the centrifugal force of any revolving body, 
expressed in pounds, is 

F = .00034 W R N 2 ; (19.) Art. 903. 

in which F = centrifugal force; 

W= total weight of body in pounds; 

R = radius, usually taken as the distance be- 
tween the center of motion and the cen- 
ter of gravity of the revolving body, in 
feet ; 

N = number of revolutions per minute. 



92 TABLES AND FORMULAS. 

THE CENTER OF GRAVITY OF TWO BODIES. 

Let / = the distance between the centers of the bodies; 
l x = the short arm ; 
w = weight of small body; 
W— weight of large body. 

l > = w£j? (20') Art. 911. 

THE EFFICIENCY OF A MACHINE. 

Let F = the force applied to the machine; 
V — the velocity ratio of the machine ; 
W= the weight actually lifted or equivalent resist- 
ance overcome; 
E = the efficiency of the machine ; 

W 
Then, E = -^y. (22.) Art. 950. 



WORK. 

If the force necessary to overcome the resistance be repre- 
sented by F, the space through which the resistance acts by 
S, and the work done by U y then U =■ F S. 

HW= the weight of a body, and h == the height through 
which it is raised, U= W h. Hence the work done 

[J=FS= IV h. (23.) Art. 953. 



POWER. 

The power of a machine may always be determined by 
dividing the work done in foot-pounds by the time in minutes 
required to do the work; i. e. , 

FS 
Power = -y- m (24.) Art. 954. 



KINETIC ENERGY. 

Let IV = the weight of the body in pounds; 
v = its velocity in feet per second ; 
h — the height in feet through which the body must 
fall to produce the velocity V; 

m — the mass of the body = — . (See formula 10.) 



TABLES AND FORMULAS. 93 

The work necessary to raise a body through a height h is 
Wh. The velocity produced in falling a height h is 



v = 4/2 g~h, and h = -—. (See formulas 15 and 16.) 

2g ' 

v 2 W 

Therefore, work = Wh = W —- = i X — X v 2 = lmv 2 , 

%g g 

or Wh = i m v\ (25.) Art. 957. 



DENSITY. 

The density of a body is its mass divided by its volume 
in cubic feet. 

Let D be the density; then the density of a body is 

D = %. Since m = —,£>= -^. (26.) Art. 962. 

V g> g V 



RULES AND FORMULAS USED IN HYDRAULICS. 



PASCAL'S LAW. 

Rule. — The pressure per unit of area exerted anywhere 
upon a mass of liquid is transmitted undiminished in all 
directions, and acts with the same force upon all surfaces in 
a direction at right angles to those surfaces. Art. 968. 



THE GENERAL LAW FOR THE DOWNWARD PRESSURE 
UPON THE BOTTOM OF ANY VESSEL. 

Rule. — The pressure upon the bottom of a vessel containing 
a fluid is independent of the shape of the vessel, and is equal 
to the weight of a prism of the fluid whose base has the same 
area as the bottom of the vessel, and whose altitude is the 
distance between the bottom and the upper surface of t lie fluid 
plus the pressure per unit of area upon the upper surface of 
the fluid, multiplied by the area of the bottom of the vessel. 
Art. 971. 

GENERAL LAW FOR UPWARD PRESSURE. 

Rule. — The upward pressure on any submerged horizontal 

surface equals the weight of a prism of the liquid whose 
base has an area equal to the area of the submerged surface, 



94 TABLES AND FORMULAS. 

and whose altitude is the distance between the submerged 
surface and the upper surface of the liquid plus t lie pressure 
per unit of area on the upper surface of the fluid, multiplied 
by the area of the submerged surface. Art. 973. 



GENERAL LAW FOR LATERAL PRESSURE. 

Rule. — The pressure upon any vertical surface due to the 
weight of a liquid is equal to the weight of a prism of the 
liquid whose base has the same area as the vertical surface, 
and whose altitude is the depth of the center of gravity of 
the vertical surface below the level of the liquid. 

Any additional pressure is to be added, as in the previous 
cases. Art. 975. 

GENERAL LAW FOR PRESSURE. 

Rule. — The pressure exerted by a fluid in any direction 
upon any surface is equal to the weight of a prism of the 
fluid whose base is the projection of the surface at right 
angles to the direction considered, and whose height is the 
depth of the center of gravity of the surface below the level 
of the liquid. Art. 979. 



SPECIFIC GRAVITY. 

Let IV be the weight of the solid in air and IV' the weight 
in water; then, IV — JJ Tf = the weight of a volume of water 
equal to the volume of the solid, and 

Sp. Gr. = -jyT-^-jp- (27.) Art. 982. 

If the body be lighter than water, a piece of iron or other 
heavy substance must be attached to it sufficiently heavy to 
sink both. Then weigh both bodies in air and both in water. 

Let W = weight of both bodies in air; 

W = weight of both bodies in water; 
w = weight of light body in air ; 
W x — weight of heavy body in air; 
W^ = weight of heavy body in water. 



TABLES AND FORMULAS. 95 

Then, the specific gravity of the light body is given by 

sp. Gr. = ^-w'y^lw-wy <yZ7a ' ) Art " 983 * 

To find the specific gravity of a liquid: 

Weigh an empty flask ; fill it wit/i water, then weigh it, 
and find the difference between the two results; this zvill 
equal the weight of the water. Then weigh the flask filled 
with t lie liquid, and subtract the weight of the flask ; the 
result is the weight of a volume of the liquid equal to the 
volume of the water. The weight of the liquid divided by 
the weight of the water is the specific gravity of ' t die liquid. 

Let W = the weight of the flask and liquid; 
W = the weight of the flask and water ; 
w = the weight of the flask. 

Then, Sp. Gr. = |^— — . {27b.) Art. 984. 

r W — w v ' 



FORMULAS FOR FLOW OF WATER. 



MEAN VELOCITY. 

Let Q — the quantity in cubic feet which passes any sec- 
tion in 1 second ; 

A = the area of the section in square feet ; 

v m — tne mean velocity in feet per second. 
Then, Q = A v m , (Z&a.) Art. 989. 

and v » l= §- (28£.) Art. 989. 



VELOCITY OF EFFLUX FROM AN ORIFICE. 

Let v = the velocity of efflux in feet per second; 
h = the head in feet on the orifice considered ; 
h 1 = the head equivalent to a pressure/; 
IV = the weight of the water in pounds flowing- 
through the aperture per second. 

U'v 2 
The kinetic energy of the issuing water = — . 

< g 



96 TABLES AND FORMULAS, 



The work the issuing water can do — ]Vh. 



Wv* 



W h — — — , or v = \'%gh. 
2g ' 



434 



, where Ji x is in feet, and p in pounds per 
square inch. 



P 
h x = ■£-=, where /i 1 is in feet, and / in pounds per 

square foot. 
h -J- h x = the total head. 



v=Wg{h x +h). (29.) Art. 991. 

If a is the area of a large orifice in the bottom of a small 
vessel whose area is A, the velocity is 

(31.) Art. 993. 




THEORETICAL RANGE OF A JET. 

Let // = head on center of orifice; 

y = vertical height of orifice above the surface where 

the water strikes; 
R = range. 

Then, R = \/~±JTy. (30.) Art. 992. 



FLOW THROUGH ORIFICES. 

Velocity of the Jet. 

Let v = theoretical velocity ; 

v' = actual maximum velocity ; 

c' = coefficient of velocity ; 

k = head on center of orifice ; 

g — acceleration due to gravity = 32.16. 

7/ = c' v = c \f%gh. (32.) Art. 994. 

An average value of c' is .98. 



TABLES AND FORMULAS. 9? 

Discharge of an Orifice. 

Let Q = theoretical discharge; 
Q = actual discharge; 
a = area of orifice; 
c" = coefficient of discharge; 
h = head on center of orifice ; 
g = acceleration due to gravity = 32.16. 

An average value of c" is .61. Then, 



Q = c" Q = c" a \/¥gh = . 01 a V^gk. (33.) Art. 994. 

Discharge of Standard Orifices. 

Let Q = discharge in cubic feet per second ; 

d = diameter of a circular or length of a side of 

a square orifice in feet; 
d' = depth of a rectangular orifice in feet; 
b = breadth of a rectangular orifice in feet ; 
// = head on the center of a circular or of a square 

orifice in feet; 
h x = head on the upper edge of a rectangular orifice 

in feet ; 
// 2 = head on the lower edge of a rectangular orifice 

in feet ; 
,c = coefficient of discharge (see tables of Coefficients 

of Discharge for Standard Orifices) ; 
g = acceleration due to gravity = 32.16. 

For a circular vertical orifice, 



Q = . 7854 d*c y%gk- 6.299 d 2 c \/Ji. (34«.) Art. 996. 
For a square vertical orifice, 



Q = cd 2 i/%g/i = $.0Zcd*i//i. (34/^.) Art. 997. 

For a rectangular vertical orifice, 

Q = cxi& V%g WK - VK) = 

5.mcb{\/7^-Vh?). (34r.) Art. 99S. 

If the head // on the center of a rectangular vertical 



98 TABLES AXD FORMULAS. 

orifice is greater than 4 d, the discharge may be computed 
by the formula 

Q = c b d\ 2g7i = 8.02 c b d \ 7i. (34^. ) Art. 998. 

For approximate computations the value of c that may be 
ased in formulas 34<r and 34** is c = .615. 
Discharge of a Submerged Rectangular Orifice. 

Let Q = discharge in cubic feet per second ; 
b = breadth of orifice in feet ; 
d = depth of orifice in feet ; 
h Q = the difference in the level of the water on the 

two sides of the orifice in feet ; 
g = acceleration due to gravity = 32.16. 
Then, 
Q = . 615 b d\ %gh = 4. 032 b d\ T Q . (34c. ) Art. 999. 



THE DISCHARGE OF WEIRS. 

Let / = length of the weir in feet ; 
H = measured head in feet ; 
v — velocity with which the water approaches the 

weir in feet per second; 
h = head equivalent to the velocity with which the 

water approaches the weir in feet ; 
c = coefficient of discharge (see tables of Coefficients 

of Discharge for Weirs) ; 
Q = theoretical discharge in cubic feet per second ; 
Q' = actual discharge in cubic feet per second. 

The theoretical discharge per second is 

Q='i\TgI(H-\-hf. (35*.~) Art. 1006. 

If there is no velocity of approach, this becomes 

Q = \s/%glH*. (35*.) Art. 1006. 

The actual discharge for weirs without end contractions i^" 
given by the formulas 

Q = c (tV*g) l {H+\hf = 5.347 cl(H+\ h)\ 

(36*. ) Art. 1006. 
and 

Q , = c(iVfg) IH 1 = 5.34:1c I H*. (36*.) Art. 1006. 



TABLES AND FORMULAS. 99 

For weirs with end contractions, the formulas are 

Q = c%\/^l(H +lAhf = 5.UV c l(H +lAhf, 

(37 a.) Art. 1006. 

and Q' = c %+/%]? /H*= 5.34:7 c I H*. (37b.) Art. 1006. 

The velocity of approach is the mean velocity with 
which the water flows through the canal leading to the 
weir. If A is the area of the cross-section of the water in 

this canal, we have v = —r, from which we see that Q' must 

A 

be determined approximately by assuming v = 0, and then 
use this value of Q' to find v. V may also be measured 
approximately by means of a float on the water in the canal 
or stream. 

r-fi 

Having found v, we have the equivalent head h = — = 

%g 

.01555 z'\ (See Arts. 990 and 991.) Since v is small with 
a properly constructed weir, it is usually neglected, unless 
great accuracy is required. 



FLOW OF WATER THROUGH F»IF»ES. 

Let / = length of pipe in feet; 

d = diameter of pipe in feet; 

d l = diameter of pipe in inches; 

v = mean velocity of flow through pipe in feet per 
second ; 

h = total head on outlet end of pipe in feet ; 

h" — head in feet equivalent to the velocity v\ 

h'"— head in feet equivalent to the loss of pressure 
at entrance to pipe; 

1i lx = head in feet equivalent to the loss in pressure 
produced by friction in pipe ; 

/i v =- head in feet equivalent to loss in pressure pro- 
duced by angular bends in pipe ; 

k vl = head in feet equivalent to loss in pressure pro- 
duced by circular bends in pipe; 

f = a coefficient for loss of head due to friction (see 
table of Coefficients /"for Smooth Iron Pipes); 

m = a coefficient for loss of head at entrance; 

n — number of bends in pipe ; 



100 TABLES AND FORMULAS. 

c = a coefficient for loss of head due to angular 

bends (see table of Coefficients for Angular 

Bends) ; 
c' — a. coefficient for loss of head due to circular 

bends (see table of Coefficients for Circular 

Bends) ; 
Q = quantity discharged by pipe in cubic feet per 

second; 
Q = quantity discharged by pipe in gallons per 

second ; 
r = radius of pipe in feet ; 
R = radius of circular bend in pipe in feet ; 
a° = number of degrees of angular bend in pipe. 

General Formulas. 

Loss of head at entrance, 

h'" = mh" = m^-. (39.) Art. 1020. 

2g 

Loss of head due to friction, 



k iy =/~^-. (40a.) Art. 1021. 



Loss of head due to angular bends, 

h v = c^-. (40b.) Art. 1023. 

Loss of head due to circular bends, 



/i yi = c'—. {40c) Art. 1023. 

Total head, 

h = h" + /;'" + // IV + m = 

^'+f-i*r- + "*■£- + *''£-. (41a.) Art. 1024. 

%g ' J d%g ' %g 2g 

Velocity of flow, 



_2g/i 
8.02 / . k . (42.) Art. 1G24. 



1 +/_ + /«+ nc 



j/ 1+fj + m + nc' 



TABLES AND FORMULAS. 101 

If m = .5 and there are no sharp bends, 



/ 



' lgh =8.03 / -■ (43.) Art. 1024. 

1.5 +/ 3 /l-«+/ 2 



and, when the diameter is in inches, 



» = 2. 315 j/ , 7/ f' , . (43a. ) Art. 1 025. 

Velocity Through Long Pipes. 

When the diameter is in feet, 

v=%MY^. (44.) Art. 1025. 

When the diameter is in inches. 



» = 2.315y^-. {44a.) Art. 1025. 

Head Required to Produce a Given Velocity, 

General formula, 



^(l+ft+m + nc'} 



& = —±- -777^- -■ (45.) Art. 1026. 

When the influence of bends is neglected and m has the 
value .5, the formula is 

h = -fj^d + • ° 233 " 2 - (45 "' ) Art - X ° 26# 



When the diameter is given in inches, 

fl* 
5.36 d 



h = {["f. + .0233 v\ [46b.) Art. 1Q26. 



The Quantity Discharged from Pipes. 

When the diameter is given in feet, the discharge in cubic 
feet per second is 

g=. 7854 <*'«>. (46.) Art. 1027. 

Since one cubic foot contains 7-48 gallons, if the diameter 
is in feet, we have 



102 TABLES AND FORMULAS. 

Q = . 785-1 d 2 v x 7. 48 gallons per second ; (46#. ) 

Art. 1027. 
and for the diameter in inches, 

<2' = .0408 < 2 z; gallons per second. (46£.) Art. 1027. 

The Diameter of Pipes. 

With k, /, and ^ in feet and the quantity Q in cubic feet 
per second, the formula for the diameter of a pipe without 
sharp bends is 

d= 0.479 [(1.5d-\-fl)^Y. (47.) Art. 1028. 

In using this formula, take the approximate value of f as 
.02, and compute an approximate value for d, neglecting the 
term 1.5 d in the second member of the formula. With 
this value of d, find the value of v from the formula 

v = ^ , 2 , and find the corresponding value of /"from the 

table of Coefficients for Pipes. 

Repeat the computation for d by placing the approximate 
values of d and f just found in the second member of the 
formula. One or two repetitions of this process will give a 
near approximation of d from which to select the pipe from 
the standard market sizes. 

For pipes whose length is more than 4,000 times their 
diameter, the following formula may be used: 



^=0.47 



9 (^X^)- (47^.) Art. 1028. 



FLOW OF WATER IN CONDUITS AND CHANNELS. 

Let 5" = slope of a conduit or channel ; 
]i — a given fall ; 

/ = distance in which the fall h occurs ; 
p — wetted perimeter ; 
a — area of water cross-section; 
r — hydraulic radius; 
v — mean velocity of flow; 
Q — quantity discharged; 



TABLES AND FORMULAS. 103 

c = a coefficient to be determined by Kutter's for- 
mula; 

n — coefficient of roughness to be used in Kutter's 
formula (see table of Coefficients of Roughness). 

Formula for slope, 

S = j. (48.) Art. 1032. 

Hydraulic radius, 

r = %. (49.) Art. 1032. 

P 
Discharge, 

Q=av. Art. 1032. 

Mean velocity, 

v=-cj/tS. (50.) Art. 1033. 

To find the value of c use Kutter's formula, 

c - ,-„ + /" ooimx« - (51 " ) Art 1033 ' 

.5021 + ^3 + — ^-j-= 

The value of n to be used in this formula is to be taken 
from the following table to correspond with the character 
of the channel: 

VALUES OF THE COEFFICIENT OF ROUGHNESS. 

For Use in Kutter's Formula. 

Character of Channel. Value of ;/. 

Clean, well-planed timber 009 

Clean, smooth, glazed iron and stoneware pipes 010 

Masonry smoothly plastered with cement on 

Clean, smooth cast-iron pipe , en 

Ordinary cast-iron pipe 012 

Unplaned timber 012 

Selected sewer pipes, well laid and thoroughly Hushed. .01 2 



Rough iron pipes 013 

Ordinary sewer pipes laid under usual conditions 013 

Dressed masonry and well-laid brickwork 015 



a. a. iv.— 20 



104: TABLES AND FORMULAS. 

Character of Channel. Value of ?i. 

Good rubble masonry and ordinary rough or fouled 

brickwork 017 

Coarse rubble masonry 020 

Gravel, compact and firm 020 

Earth canals, well made and in good alinement 0225 

Rivers and canals in moderately good order and per- 
fectly free from stones and weeds 025 

Rivers and canals in rather bad condition and some- 
what obstructed by stones and weeds 030 

Rivers and canals in bad condition, overgrown with 
vegetation and strewn with stones and other 
detritus, according to condition 035 to .050 



FORMULAS USED IIV PNEUMATICS. 



PRESSURE, VOLUME, DENSITY, AND WEIGHT OF AIR 
WHEN THE TEMPERATURE IS CONSTANT : 

Mariotte's Law. — The temperature remaining the same, 
the volume of a given quantity of gas varies inversely as the 
pressure. 

Let/ = pressure for one position of the piston; 

p x = pressure for any other position of the piston; 
v = volume corresponding to the pressure/; 
v 1 = volume corresponding to the pressure p x . 

Then, p v=p x v v (53.) Art. 1049. 

Let D be the density corresponding to the pressure/ and 
volume v, and D 1 be the density corresponding to the 
pressure p x and volume z\\ then, 

p\D =rp x \D» or pD 1 = p i D, (54.) Art. 1052. 

zndv:£> l = v i :D, or v D = v x D x . (55.) Art. 1052. 

Thus, let [Fbe the weight of a cubic foot of air or other gas, 
whose volume is v } and pressure is/; let Jl\ be the weight 
of a cubic foot when the volume is v v and pressure is p y \ 
then, 

P ll \=Pi W. (56.) Art. 1052. 

v W = v x W v (57.) Art. 1052- 



TABLES AND FORMULAS. 105 

PRESSURE AND VOLUME OF A GAS WITH VARIABLE 
TEMPERATURE : 

Gay-Lussac's Law. — If the pressure remains constant, 
every increase of temperature of 1° F. produces in a given 
quantity of gas an expansion of -^ Y of its volume at 32° F. 

If the pressure remains constant it will also be found that 
every decrease of temperature of 1° F. will cause a decrease 
of -fa of the volume at 32° F. 

Let v = original volume of gas ; 
v' x = final volume of gas ; 

t = temperature corresponding to volume v\ 
t x — temperature corresponding to volume v x . 

Then, Vi = v {M°±Ly (58.) Art. 1054. 

That is, the volume of gas after heating (or cooling') equals 
the original volume multiplied by IfiO plus the final tempera- 
ture divided by Jf.60 plus the original temperature. 

Let p = the original tension; 

t = the corresponding temperature; 
p x = final tension; 
t x = final temperature. 

Then, A=/ (g°±£j. (59.) Art. 1055. 

Let / = pressure in pounds per square inch ; 
V= volume of air in cubic feet; 
T= absolute temperature; 
W= weight in pounds. 

Then, p V= .37052 T. (60.) Art. 1056. 

If the weight of the air be greater or less than 1 pound, 
the following formula must be used : 

/ V=. 37052 WT. (61.) Art. 1057. 
Let p v J\, and T t represent the pressure, volume, and 
temperature of the same weight of air in another state; 
then, 

^ = ^" L . (62.) Art. 1Q58, 



106 TABLES AND FORMULAS. 

MIXTURE OF TWO GASES HAVING UNEQUAL VOLUMES 
AXD PRESSURES. 

Let v and p be the volume and pressure, respectively, of 
one of the gases. 

Let v x and p 1 be the volume and pressure, respectively, 
of the other gas. 

Let V and P be the volume and pressure, respectively, of 
the mixture. Then, if the temperature remains the same, 

VP= vp + v x p x . (63.) Art. 1G62. 



MIXTURE OF TWO VOLUMES OF AIR HAVING UNEQUAL 
PRESSURES, VOLUMES, AND TEMPERATURES. 

If a body of air having a temperature t lt a pressure p v and 
a volume i\ be mixed with another volume of air having a 
temperature /„, a pressure />.,, and a volume 7 r n , to form a 
volume V having a pressure P, and a temperature /, then, 
either the new temperature t, the new volume J\ or the new 
pressure P may be found, if the other two quantities are 
known, by the following formula, in which T yl T t1 and T 
are the absolute temperatures corresponding to t lt / 2 , and t\ 

PV=[^+^-i\T. (64.) Art 1Q63. 



FORMULAS USED IN STRENGTH OF 
MATERIALS. 



UNIT STRESS, UNIT STRAIN, AXD COEFFICIENT OF 
ELASTICITY. 

Let P— the total stress in pounds; 

A = area of cross-section in square inches ; 
5 = unit stress in pounds per square inch ; 

/ = length of body in inches; 

c = elongation in inches; 

s = unit strain ; 

E = coefficient of elasticity. 

S = ~, or P= A S. (65.) Art. 1 103. 



TABLES AND FORMULAS. 10? 

s = j, or e = I s. (66.) Art. 1 1 04. 

9 P e PI 

£=- = ^4-^- = ^-. (67.) Art. 1110. 

s A I A e 



aTRENGTH OF PIPES AND CYLINDERS. 

Let d = inside diameter of pipe in inches; 
/ = length of pipe in inches; 
p = pressure in pounds per square inck; 
P = total pressure ; then, P^ p I d\ 
t — thickness of pipe; 
5 = working strength of the material. 
For longitudinal rupture 
/ / d - 2 t I S, or 
p d=2 t S. (68.) Art. 1 1 23. 

For transverse rupture 

pd^kt S. (69.) Art. 1 1 24. 

Since, for longitudinal rupture, / d = 2 / S, it is seen that 
a cylinder is twice as strong against transverse rupture as 
against longitudinal rupture. 

For pipes and cylinders whose thickness is greater than 
T \ of the radius, use the following formula, in which 
r = the inner radius, and the other letters have the same 
meaning as before. 

/ = -^-r~, (70.) Art. 1 1 25. 

The following formula gives the collapsing pressure in lb. 
per sq. in. for wrought-iron pipe: 

/ =9,600,000^-.' (71.) Art. 1126. 



MOMENT OF INERTIA, RESISTING MOMENT, AND BENDING 
MOMENT OF BEAMS. 



Let / = moment of inertia; 
A — area of cross-section ; 
r = radius of gyration; 



108 TABLES AND FORMULAS. 

c = distance, from neutral axis to outermost, fiber ; 
5 4 = ultimate strength of flexure ; 
f = factor of safety ; 
M — bending moment. 

I=Ar\ (72.) Art. 1154. 

The resisting moment is given by the expression 

S" S" / 

±A r* = — L or S -. 

c c c 

For the bending moment 

M=sJ-, (73.) Art. 1156. 

and, when a factor of safety is used, 

^=yj- (74.) Art. 1159. 



DEFLECTION OF A BEAM. 

Let a = a constant depending on the manner of loading 
the beam and the condition of the ends; 
s = the deflection; 
E = the coefficient of elasticity; 
/ = the length of the beam in inches; 
W= the total weight supported in pounds; 
/ = moment of inertia about the neutral axis. 

s^a^j. (75.) Art. 1162. 



STRENGTH OF COLUMNS. 

Let IV = load on a column; 

S 2 = ultimate strength for compression; 

A = area of section of column ; 

f = factor of safety; 

/ = length of column in inches; 

g = a constant to be taken from table ; 

/ = least moment of inertia of cross-section; 

/; = length of longer side of a rectangular column; 



TABLES AND FORMULAS. 109 

d = length of shorter side of a rectangular column, 

or the diameter of a circular column; 
c = length of one side of a square column. 

W= / 5 ^ /2x . (76.) Art. 1169. 

'(' + 77 



For a circular column, 

^2 *\ A ^2 <£" / 

(78.) Art. 1171. 
For a rectangular column, assume d, then, 

. >(i+5f) 



^5. 



(79.) Art. 1171, 



STRENGTH OF SHAFTS. 

Let d = diameter of a round shaft, or side of a square 

shaft, in inches; 
c — a constant (see table of Constants for Shafting) ; 
c t = a constant (see table of Constants for Shafting) ; 
P= a. force applied at the end of a lever arm in 

pounds; 
r — length of lever arm in inches; 
H = horsepower transmitted by shaft; 
N = number of revolutions per minute; 
k — a constant (see table of Constants for Shafting) ; 
k x = a constant (see table of Constants for Shafting) ; 
q — a constant (see table of Constants for Shafting) ; 
q x — a constant (see table of Constants for Shafting). 

For all solid shafts below 11 inches in diameter use the 
formula 

d= c ^P7-c x ^^-. (80.) Art. 1173. 



110 



TABLES AND FORMULAS. 



If the diameter of a wr ought-iron shaft is greater than 
12.4*, of a cast-iron shaft greater than 10. 3", or of a steel 
shaft greater than 13.6", use the following formula: 



d 



H 



= k¥Pr=tk x Vj r 



(81.) Art. 1174, 



For a hollow (round) shaft use formula 82 or 83^ 

P =Q ^*J?* \ ( 82 -) Art. 1174. 

H=q x N ^~ d \ (83.) Art. 1174, 



CONSTANTS FOR SHAFTING. 



VALUES OF c AXD c, TO BE USED IX FORMULA 80. 



Material. 


c 


Cx 




Round. 


Square. 


Round. 


Square. 


Wrought Iron . 

Cast Iron 


.310 

•353 
.297 


.272 

•309 
. 260 


4.92 

5-59 
4.70 


4-3i 
4.89 
4. 11 


Steel 





VALUES OF k, k lt q, AXD q, TO BE USED IX FORMULAS 
81, 82, AXD 83. 



Material. 


k 


k, 


Q 


?i 


Wrought Iron 

Cast Iron 


.0909 

•ii45 
.0828 


3.62 

4-5 6 
3-3° 


J ,335 
669 
1,767 - 


.0212 
.0106 


Steel 


. 0280 







STRENGTH OF ROPES AXD CHAINS. 

Let P= working or safe load in pounds; 
C= circumference of rope in inches; 
d — diameter of the link of a chain in inches. 



TABLES AND FORMULAS. Ill 

For manila ropes, hemp ropes, or tarred hemp ropes, 
P=100C\ (84.) Art. 1175. 

For iron wire rope of 7 strands, 10 wires to the strand, 
P=G00C\ (85.) Art. 1176. 

For the best steel wire rope, 7 strands, 19 wires to the 
strand, 

P= 1,000 C\ (86.) Art. 1176. 

For open-link chains made from a good quality of wrought 
iron. 

P= 12,000a 72 , (87.) Art. 1179. 

and for stud-link chains, 

P = 18,000 d\ (88.) Art. 1179. 



FORMULAS USED IN SURVEYING. 



RADIUS OF A CURVE. 

To find the radius, the degree being given: 
Let R = the length of the required radius; 

P= the deflection angle equal to one-half the degree 
of the given curve. 

50 
R = -—7,. (89.) Art. 1 249. 

sin D v ' 



LENGTH OF SUB-CHORDS. 

For curves of short radii: 

Let C = the length of the required chord; 

R = the radius of the given curve; 

D = the deflection angle of the given curve, equal to 
one-half its degree. 

C = 2 R sin D. (90.) Art. 1 250. 



LENGTH OF THE TANGENT OF A CURVE. 

When the radius and intersection angle are given: 
Let T ■=- the length of the required tangent; 

R = the radius of the given curve ; 

/ = the intersection angle of the given curve. 

T=Rta.niI. (91.) Art. 1251- 






112 TABLES AND FORMULAS. 

CHORD DEFLECTION. 

When the length of the chord and the radius are given 
Let d — the required chord deflection ; 

c — the length of the chord of the given curve; 

R = the radius of the given curve. 

d= C R' (92.) Art. 1255. 



TANGENT DEFLECTION. 

When the length of the tangent, or of its corresponding 
chord, and the radius are given : 

Let c = the length of the tangent or corresponding chord, 
R = the radius of the given curve. 

tangent deflection = —=. (93.) Art. 1255. 

Or, find the chord deflection as in the preceding formula 
and divide it by 2. The quotient is the required tangent 
deflection. 

STADIA MEASUREMENTS. 

To find the horizontal distance between two given points, 
the distance between them having been read with the stadia 
and the vertical angle taken ; 

Let D = the corrected or horizontal distance; 
c = the constant; 
a k = the stadia distance ; 
11 = the vertical angle. 

D = c cos n + a k cos 2 n. (94.) Art. 1 301 . 

To find the difference of elevation between two given 
points in stadia work: 

Let E = the required difference in elevation ; 
c — the constant ; 
a k =. the stadia distance; 
n = the vertical angle. 

£ = c sin 11 + a k Sm J* " . (95.) Art. 1301. 



TABLES AND FORMULAS. 113 

BAROMETRICAL LEVELING. 

To find the difference of elevation between two points 
with the aneroid barometer: 

Let Z"= the difference of elevation between the two given 
stations ; 
h = the reading in inches of the barometer at the 

lower station; 
77= the reading in inches of the barometer at the 
higher station; 
/ and f = the temperature (F. ) of the air at the two stations. 

Z = (log h - log H) x 60, 384. 3 (l + - + 'J" 64 ° ). 

(96.) Art. 1304. 

RULES AND FORMULAS USED IN SURVEYING 
AND MAPPING. 

Rule for Balancing a Survey. — As the sum of all 

the courses is to any separate course, so is the whole difference 
in latitude to the correction for that course. A similar 
proportion corrects the departures. Art. 1315. 

Rule for Double Longitudes. — The double longitude 
of the first course is equal to its departure. 

The double longitude of the second course is equal to the 
double longitude of the first course plus the departure of that 
course plus the departure of the second course. 

The double longitude of the third course is equal to the 
double longitude of the second course plus the departure of 
that course plus the departure of the course itself. 

The double longitude of any course is equal to the double 
longitude of the preceding course plus the departure of that 
course plus the departure of the course itself. 

The double longitude of the last course (as zvell as of the 
first) is equal to its departure. This result, when obtained 
by the above rule, proves the accuracy of the calculation of the 
double longitudes of all the preceding courses. Art. 1319. 



1U TABLES AND FORMULAS. 

APPLICATION OF DOUBLE LONGITUDES TO FINDING 

AREAS. 

1. Prepare ten columns, and in the first three Write the 
stations, bearings, and distances, respectively. 

2. Find the latitudes and departures of each course by the 
Traverse Table ', placing them in the four following columns. 

3. Balance them by the above rule for balancing a survey, 
correcting them in red ink. 

4. Find the double longitudes by the rule for double longi- 
tudes, with reference to a meridian passing through the 
extreme east or west station, and place them in the eigJith 
column. 

5. Multiply the double longitude of each course by the cor- 
rected latitude for that course, placing the north products in 
the ninth column and the south products in the tenth column. 

6. Add the last two columns; stibtract the smaller sum 
from the larger, and divide the difference by 2. The quotient 
zvill be the area required. Art. 1321. 



AREAS OF IRREGULAR FIGURES. 

Trapezoidal Rule. — Divide the figure into any sufficient 
number of equal parts by means of vertical lines called 
ordinates ; add half the sum of the two end ordinates to the 
sum of all the other ordinates ; divide by the number of spaces 
{that is, by one less than the number of ordinates) to obtain 
the mean ordinate, and multiply the quotient by the length of 
the section to obtain the area. 

Simpson's Rule. — Divide the length of the figure into 
any even number of equal parts, at the common distance D 
apart, and draw ordinates through the points of division. 
Add together the length of the first and the last ordinates 
and call the sum A ; add together the even ordinates and call 
the sum B ; add together the odd ordinates, except the first 
and the last, and call the sum C. 

Then, area of figure = * + 4 f + 2 C X D. Art. 1324. 



TABLES AND FORMULAS. 115 

VOLUMES OF IRREGULAR SOLIDS. 

To find the volume included between two parallel cross- 
sections whose areas are known, 

Let A = area of one section in square feet; 

B = area of the other section in square feet; 

C= distance between the two sections in feet; 

D — required volume in cubic feet. 
Then, approximately, 

D = A + B X C. (97.) Art. 1325. 

The Prismoidal Formula. — A more accurate result 
than that given by the last formula is given by the 
prismoidal formula. 

Let A = area of one section in square feet; 

B = area of the other section in square feet; 

M= the area of the average or mean section in 

square feet; 
L = distance between the sections in feet; 
5 = the required volume in cubic feet. 

S = ^(A + ±M+B). (98.) Art. 1326. 



LATITUDES AND DEPARTURES. 

To find the latitude and departure of a course by means 
of a table of sines and cosines, 

Latitude = distance X cosine bearing. (99.) Art. 1338* 

Departure = distance X sin bearing. (lOO.) Art. 1338. 



FORMULAS USED IN STEAM AND STEAM 
ENGINES. 



SPECIFIC HEAT. 

W= weight of body in pounds; 
/ = temperature before heat is applied; 
/, = temperature after heat is applied; 
c = specific heat of body ; 



116 TABLES AND FORMULAS. 

U — number of B. T. U. required to raise temperature of 
body from t to t x . 

U= c JV(t i - t). (lOl.) Art. 1379. 



TEMPERATURE OF MIXTURES. 

zv, w l7 ze/ 2 , . . . .= weights of the several substances, respect- 
ively; 
c, c 19 c %i . . ■. . = specific heats of the substances, respect- 
ively; 
t y t lf t 99 . . . . = temperatures of the substances, respect- 
ively ; 
T — final temperature of mixture. 

r= ^^ + ^ 1 r 1 / 1 + ^r,/, + ...- > Art . 1383 . 

W C + W 1 C x + 7t' 2 C 2 + . . , . 

Mixture of Steam and Water. 

W = weight of steam in pounds; 

w = weight of water in pounds; 

t x — temperature of steam; 

/ = temperature of water; 

T = final temperature of mixture; 

L = latent heat of steam at the given temperature. 

r= U '' (L + t ' ) + *" . (102.) Art. 1384. 

/ / -4- W 



WORK DONE BY PISTON. 

p = net pressure on piston in pounds per square inch; 
V = volume in cubic feet swept through by piston ; 
}V — work done by moving piston. 

JV= lUp V. (103.) Art. 1395. 



REAL AXD APPARENT CUT-OFF. 

s = apparent cut-off; 

k — real cut-off; 

i — clearance expressed as a per cent, of the stroke. 

j. s + * 



l+'i 



(104.) Art. 1457. 



TABLES AND FORMULAS. 117 

HORSEPOWER. 

I. H. P. = indicated horsepower of engine; 

P^=± mean effective pressure in pounds per sq. in. ; 

A = area of piston in square inches; 

L = length of stroke in feet; 

IV = number of strokes per minute. 

PT A N 
L H - P - = -3P00- (1 ° 5 - ) Art 1 495 - 

MEAN EFFECTIVE PRESSURE. 

p = gauge pressure ; 

k = constant depending upon cut-off (see table of 
Constants used in determining M. E. P.); 
M. E. P. = mean effective pressure. 

M. E. P. = .9 [k(p+ 14.7) -17]. Art. 1496. 



PISTON SPEED. 

/ = length of stroke in inches; 
R = number of revolutions per minute; 
5 = piston speed in feet per minute. 

S = 1 ^. (106.) Art. 1497. 



MECHANICAL EFFICIENCY OF ENGINE. 

I. H. P. = indicated horsepower; 

Friction H. P. = horsepower absorbed in overcoming fric- 
tion of engine; 

Net H. P."= I. H. P. —Friction H. P. = horsepower avail- 
able to perform useful work ; 

E = efficiency of engine. 

£ = SoT" Art!. 1499. 



STEAM CONSUMPTION. 

/ = distance between two points on the indicator dia- 
gram, one on the expansion line, and the other on 
the compression line, both being equally distant 
from the vacuum line: 



118 TABLES AND FORMULAS. 

L = length of indicator diagram; 

a = absolute pressure of steam at the two points chosen 

IV= weight of a cubic foot at pressure a ; 

Q = steam consumption in pounds per I. H. P. per hour. 

g= i8,wo/y (107>) Art 1507 _ 



THERMAL EFFICIENCY OF ENGINE. 

7,= absolute temperature of steam entering cylinder 
7„ = absolute temperature of steam leaving cylinder; 
E = thermal efficiency. 



7, 



Art 1512. 



WATER REQUIRED BY CONDENSER. 

t 1 = temperature of departing condensing water; 

t 2 — temperature of entering condensing water; 

t % — temperature of the condensed steam upon leaving 

the condenser; 
H = total heat of vaporization of one pound of steam at the 

pressure of the exhaust (see steam table, column o) ; 
JV = weight of water required to condense a pound of steam 

/r^ ^T^t 32 - (108.) Art. 1520. 



RATIO OF EXPANSION. 

e = ratio of expansion in high-pressure cylinder; 
E — total ratio of expansion; 

v — volume of cylinder receiving steam from boiler; 
V= volume of cylinder or cylinders exhausting into 
atmosphere or condenser. 

E = -—. (109.) Art. 1527. 



TABLES AND FORMULAS. 119 

FORMULAS USED IN STEAM BOILERS. 



AIR REQUIRED FOR COMBUSTION AND HEAT OF 
COMBUSTION. 

C = percentage of carbon in a fuel expressed in parts of 

a hundred; 
H = percentage of hydrogen in a fuel expressed in parts 

of a hundred ; 
A = cubic feet of air required to burn a pound of the fuel. 

A = 1.5%{C+3IQ. (110.) Art. 1546. 

B — British thermal units produced by the combustion of 

the fuel; 
W = weight of water that can be evaporated by a pound 

of the fuel. 

B = 145 C + 620 77. (111.) Art. 1547. 

W=~. Art. 1547. 

yob 

STRENGTH OF BOILER SHELLS. 

P = gauge pressure of steam, pounds per square inch ; 

d = diameter of shell in inches; 

/ = length of shell in inches; 

t = thickness of material ; 

S = safe stress in material: 9,000 lb. for wrought iron; 

11,000 lb. for steel; 
F — total force tending to rupture the shell; 
e = efficiency of joint (see table of Riveted Joints). 

F=Pdl. (112.) Art. 1603. 

p= %S_t_e ( 113 .) Art . igo4. 



HORSEPOWER OF BOILERS. 

W= pounds of water evaporated per hour; 
F = factor of evaporation (see table of Factors of Evapo- 
ration) ; 
H = horsepower of boiler. 

IV F 
H=--—. (114.) Art. 1618. 

o-±. o 

a. o. iv.— so 



120 TABLES AND FORMULAS. 

THE SAFETY VALVE. 

A = area of opening in valve-seat in square inches; 

p = blow-off pressure of valve; 

a = power arm of lever valve; i.e., the distance of valve- 
stem from fulcrum; 

d = weight arm of lever valve; i.e., the distance of 
weight from fulcrum; 

H = reading of spring scale, when the lever and valve are 
attached to it, at the point where the valve-stem 
joins the lever; 

P — weight of ball hung on end of lever ; 

W— weight required in a dead-weight valve; 

5 = pounds of steam generated per hour. 

W=pA. (115.) Art. 1621. 

/ = —. (116.) Art. 1621. 

pa A =Pd. Art. 1 623. 

(pA -H)a = Pd. j 
d= (pA -H)a I (117.) Art. 1624 

^ = :^44rV (US-) Art. 1627. 
p-j-10 



DRAFT PRESSURE OF CHIMNEY. 

H = height of chimney in feet ; 

T a — absolute temperature of air; 

T e = absolute temperature of escaping gases; 

p = draft pressure in inches of water. 



P 



-*fe-x> 



/ (119.) Art 1662. 



/7.6 _ 7.9\' 



TABLES AND FORMULAS. 121 

QUALITY OF STEAM (BARREL CALORIMETER), 

IV = weight of cold water in barrel; 

w — weight of mixture of steam and water run into 

barrel; 
t = temperature of steam corresponding to observed 

pressure; 
t i = original temperature of cold water; 
t 2 = temperature of cold water after steam is condensed; 
L = latent heat of a pound of steam at the observed 

pressure (see column 4, steam tables) ; 

x = portion of weight w that is dry steam ; 

x 
Q = quality of steam = — . 

W L Vw J Art. 1714. 



FORMULAS USED IN WATER-WHEELS. 



THEORETICAL ENERGY OF A GIVEN HEAD AND WEIGHT 

OF WATER. 

Let h — available head ; 

v = velocity the water would attain if it fell freely 

through the height h ; 
W= weight of water; 

g = acceleration due to force of gravity = 32.16; 
K = theoretical energy. 

K= Wh=. w£-. (121.) Art. 1727. 

%g 



THEORETICAL POWER. 

Rule. — To find the theoretical horsepower that a given 
quantity of water will furnish, multiply the weight of water 
that falls in one second by the distance through which it falls, 
and divide this product by 560 ; the quotient will be the 
theoretical horsepower. 



122 TABLES AND FORMULAS. 

Let H. P. = theoretical horsepower; 

Q — quantity of water falling in cubic feet per 

second ; 
H = total available fall in feet. 

H. P. = Q X ™' 5 X H = .1136 QH. (122.) Art. 1730. 



ENERGY OF A JET. 

Let K = energy of the jet; 

IV — weight of water that flows from the orifice or 

nozzle in one second ; 
w — weight of a cubic foot of water = 62.5 pounds; 
a = area of the jet in square feet; 
v = velocity of flow from the orifice in feet per 

second; 
c = coefficient of velocity for the orifice; 
h = head on the orifice in feet ; 
g = acceleration due to gravity = 32.16. 

K = W^~ = c* Wh. ( 1 23.) Art. 1731. 

W=wav. (124.) Art. 1731. 

K='^- = c*wavk (125.) Art. 1731, 



PRESSURE DUE TO IMPULSE AND REACTION OF A JET. 

Let P — pressure produced by the impulse-, 
R = reaction of the jet; 
IV = weight of water that flows from the orifice or 

nozzle in one second ; 
w— weight of a cubic foot of water = 62.5 pounds; 
a = area of the jet in square feet; 
•u = velocity of flow from the orifice in feet per 

second ; 
c = coefficient of velocity for the orifice; 
h = head on the orifice in feet; 
g — acceleration due to gravity = 32.16. 



TABLES AND FORMULAS. 123 

Pressure on a Vertical Surface. — When the jet 
impinges on a vertical surface the pressure is 

P=zwa— = 2c*wa& = W-. (126.) Art. 1732. 

g g 

Reaction. — The reaction of the jet is 

R = P=wa— = 2c 2 wah = W-. (127.) Art. 1732. 

g g 

Pressure Produced by Change of Direction. 

Let a° = the angle between the original direction of the 
jet and its direction after being deflected. 

The pressure exerted on the deflecting surface in the 
original direction of the jet is 

P=(l- cos a°) W-. (128.) Art. 1734. 

Pressure on a Hemispherical Cup. — When the jet 
strikes into a hemispherical cup a° = 180°, and the pressure is 

P=(l- cos 180°) W- = 2 W-; 
g g 

that is, the pressure is twice as great as the, pressure pro- 
duced when the jet struck a surface at right angles to its 
direction of motion. Art. 1734. 

Effect When the Surface Is in Motion. 

Let v' = the velocity with which the surface moves along 
the line of motion of the jet. 

The pressure on the surface is 
P=(l-cosa°)w(^l-~y^^-. (129.) Art. 1735. 

If the surface is a hemispherical cup, the pressure is 

IV 
P=.0622—(v-v'y. (130.) Art. 1735. 

The theoretical work of a jet impinging in a moving 
hemispherical cup is a maximum when the velocity if the cup 



124 TABLES AND FORMULAS. 

is one-half the velocity of the jet, and it is equal to the 
theoretical work that would be done by the energy due to the 
velocity of the water. Art. 1 735. 



EFFICIENCY. 

Rule I. — To find the amount of work or power that can 
be obtained from a given fall of water when the efficiency of 
the motor is given, multiply the theoretical work or power by 
the efficiency expressed as a decimal fraction, and the product 
will give the available work or pozver. Art. 1737. 

Rule II. — To find the quantity of water required to fur- 
nish a given amount of power with a given efficiency, divide 
the theoretical quantity of water by the efficiency; the quotient 
will be the quantity required. Art. 1 737. 



OVERSHOT WATER-WHEELS. 

Let H = total fall of water ; 

v = velocity of circumference of wheel; 

v e = velocity with which water enters wheel ; 

h — head required to produce velocity of entry v e ; 

D = outside diameter of wheel ; 

N '= number of revolutions of wheel; 

Z = number of buckets; 

b = breadth of buckets; 

d = depth of buckets; 

Q — quantity of water in cubic feet per second : 

c = clearance between wheel and trough. 

v e = H v to 2 v. (131.) Art. 1743. 

h = l.lj£. (132.) Art. 1743. 

2g J 

D=H-(h + c). Art. 1 743. 

^=19.1-^. (133.) Art. 1743. 

Z=nDto3D. (134.) Art. 1743. 

d= 10 inches to 15 inches. (135.) Art. 1743. 

£=3-^-to4-p-. (136.) Art. 1743. 

dv dv v ' 



TABLES AND FORMULAS. 125 

BREAST WHEELS. 

The following rules may be used for the principal dimen- 
sions of a breast wheel: 

Velocity of circumference of wheel v = 2 feet per second 
to 8 feet per second. Velocity of entry v t = \\v to %v. 

Depth of floats d = 10 inches to 15 inches. Pitch of 
floats t — d. 

Diameter of wheel, about twice the total head. 

Breadth of wheel, b = \h-4- to 2-iP-, where is in cubic 

1 dv dv 

feet per second, b and d in feet, and v in feet per second. 

Art. 1749. 

UNDERSHOT WHEELS. 

Let H. P. •= horsepower; 

v = velocity of water in race in feet per second; 
v 1 = velocity of circumference of wheel in feet per 

second; 
Q = quantity of water flowing through race in 

cubic feet per second; 
F = area of the immersed portion of the float of a 

paddle wheel in an unconfined current. 

For a wheel in a confined race, 

H. P. = .0012 (v-7'J v x Q. (137.) Art. 1754. 

For a simple paddle wheel in an unconfined current, 
H. P. = .001 (v - v x ) v v x F. (138.) Art. 1 755. 



PONCELET'S WHEEL. 

Let H = total fall in feet; 

Q = the quantity of water in feet per second ; 

D = the outside diameter of the wheel in feet; 

d = depth of floats in feet ; 

d 1 — depth of water current entering the wheel in feet ; 

v e = the velocity of the water current entering the 

wheel in feet per second ; 
b = breadth of the wheel and of the sluice in feet : 



126 



TABLES AND FORMULAS. 



7\ = velocity of circumference of wheel in feet per 

second ; 
R = radius of curvature of floats; 
A = angle A OB (see Fig. 527, Art. 1756); 
u = number of revolutions per minute of wheel ; 
n — number of floats in wheel. 

H and Q must be determined by actual measurement ; 
the other dimensions may then be made as follows: 

D = 3* ff; 

d =\D to i D; 
d^=-\. foot to 1 foot ; 
7 / e = 7.G2V1; 



b = ^' 

R = lld; 

a = 20° to 45° 

u = 19.1 7J 



D 9 



n 



D. 



Art. 1757, 



TURBINES. 

In the rules and formulas used to determine the principal 
dimensions of reaction turbines 

Let Q — the available quantity of water in cubic feet per 

second ; 
h — the total available head on the wheel in feet ; 
v e = the velocity of the flow from the guide buckets 

in feet per second; 
v r = the relative velocity of water entering the wheel 

buckets in feet per second; 
v 2 = the relative velocity of flow from the wheel 

buckets in feet per second; 
v f = the absolute velocity of the water leaving the 

wheel buckets; 
v ir = the velocity of the wheel buckets at entrance; 
v' w = the velocity of the wheel buckets at discharge; 



TABLES AND FORMULAS. 12? 

a = the angle which the direction of outflow from 

the guides makes with the radius in a radial- 
flow turbine or with a perpendicular to the 

direction of motion of the wheel buckets in an 

axial-flow turbine; 
a = the angle which the relative direction of inflow 

to the wheel makes with the same lines; 
<? 2 = the angle which the relative direction of flow 

from the vanes makes with the same lines; 
A = the effective outflow area of guide passages in 

square feet ; 
A 1 = the effective inflow area of wheel passages in 

square feet.; 
A 2 = the effective outflow area of wheel passages in 

square feet; 
A 3 == sectional area of flow for draft tube in square 

feet; 
A t = effective outflow area of draft tube in square 

feet; 
N = the number of revolutions per minute; 
r = the mean radius of an axial turbine in feet ; 
r x = the radius of the wheel at inflow, in feet, for a 

radial-flow turbine; 
r 2 — the radius of the wheel at outflow, in feet, for a 

radial-flow turbine; 
g = acceleration due to force of gravity; 
k = a coefficient for finding the radius r or r , from 

the area A ; 
AT, = a coefficient for finding the velocity^; 
P = the pitch of the guide buckets; 
P x = the pitch of the wheel buckets; 
Z = the number of guide buckets; 
Z x — the number of wheel buckets; 
/i o = the height of guide buckets in an axial-flow 

turbine ; 
h\ = the height of wheel buckets in an axial-flow 

turbine ; 



128 TABLES AND FORMULAS. , 

x — the distance between the outflow ends of two 
consecutive guide buckets, measured perpen- 
dicular to the direction of flow ; 

x % = the distance between the outflow ends of two 
consecutive wheel buckets, measured perpen- 
dicular to the direction of flow; 

/ = the thickness of guide buckets near ends; 

t 1 = the thickness of wheel buckets near ends; 

s = the part of the distance x that would be covered 
by the inflow end of one wheel bucket, to be 
measured in the same direction as x; 

e = the width of outflow ends of guide buckets; 

e A = the width of inflow ends of wheel buckets; 

r 2 = the width of outflow ends of wheel buckets. 

General Relations. — The usual proportions and values 
to be used in designing the different types of wheel are as 
follows : 

(a) For axial turbines using a large quantity of water 

under a low head, where — is greater than 16 square feet, 

a = 70° to 66°. 
2 = 70° to 66°. 
^=.67. 

k = 1 to \\. 

P — 10 inches to 12 inches. 

j \ inch to f inch for cast iron. 

I \ inch to f inch for wrought iron. 

Ji — h — - to-, 
o 6 

(b) For axial turbines using a medium quantity of water 
under medium head, where — is greater than 2 and less 
than 16 square feet, 

a = 75° to 70°. P= ^— to £-. 

a 2 = 74i-° to 73°. , ' J ' ( ° . 

' t = t = same as above. 

A, = . < . 

T T 

k =1.25 to 1.5. h = /i = — to — -. 

•i i. 



TABLES AND FORMULAS. 129 

(V) For axial turbines using a small quantity of water 
under a high head, where — is less than 2 square feet, 

a = 75° to 73°. p = 44, to 6 inches. 

a, = 77 to 74 . t = t x = same as above. 

K x = . 07. r r 

*= 1.5 to a. h = h °=%h to z 

(</) For radial inward-flow turbines, where 2 ranges 
from 2.4 to 275 cubic feet per second, and h is from 3 feet 

to 80 feet. 

a = 80° to 66°. 
a 2 = 80° to 66°. 
r„ = t} r, to -jj- r x . 
K x — 0.725 to 0.64. 
k = 0.75 to 1.75. 
P = 4J inches to 12 inches. 
t = t x = same as for axial-flow turbines. 
Z X = Z to .7 Z. 

(r) For radial outward-flow turbines, where 2 ranges 
from 2.5 to 350 cubic feet per second, and h ranges from 
3 feet to 25 feet, 

a = 75° to 66° and less. 
a 2 = 80° to 60° and less. 
?\ = i r x to \ r x . 
K x = 0.725 to 0.64. 
k = 1.5 to 2. 

*=&*>*■ 

t = t x = same as for axial-flow turbines. 
Z x ' = 1. 2 Z to 1. 3 Z. Art. 1 782. 

Velocity of Entrance. — From the general relations 
select a value of K x to correspond with the type of wheel 
and the conditions under which it works; then, 



= K l \/%gh. (139.) Art. 1783. 



130 TABLES AND FORMULAS. 

Effective Area. — From and v c the effective area A 
of the passages from the guide buckets is computed from 
the formula 

A=@-. .(140.) Art. 1783. 

v e 

Radius. — From this value of A the mean radius of a 
parallel-flow wheel is computed from the formula 

r = k</A, (141a.) Art. 1783. 

where k is a coefficient that depends on the relation between 
Q and h. (See general relations.) 

For a radial-flow turbine, the radius of the wheel where 
the water enters is given by the formula 

r 2 = k^A. (141/0 Art. 1783. 

where k depends on the style of wheel, whether outward 
flow or inward flow. (See general relations.) 

Revolutions. — The number of revolutions per minute 
is given by 

N= \ X Q ° = 9.549^, (142«.) Art. 1783. 

for axial-flow turbines, and for radial-flow turbines, 
N = 9.5-49^. (142/^.) Art. 1783. 

Number of Vanes. — Having chosen the pitch P approxi- 
mately to suit the given conditions, the number of guide 
vanes for an axial-flow turbine is given by the formula 

Z=^f, (143a.) Art. 1786. 

and for a radial-flow turbine the number of guide vanes is 

Z=^~^. (143/^.) Art. 1786. 

These formulas give approximate values for Z, and the 
actual value is the nearest corresponding whole number. 
The number of wheel vanes Z x for axial-flow turbines 



TABLES AND FORMULAS. 131 

should always be greater than Z. For ordinary cases we 
may take 

Z 1 = Z+2. (144fl.) Art. 1787. 

For radial inward-flow turbines use the values 

Z 1 = Zto'.7'Z, (144*.) Art. 1787. 

and for radial outward-flow turbines 

Z 1 = l.'2>Ztol.SZ. (144*0 Art. 1787. 

Pitch. — The exact pitch for the guide vanes of axial-flow 
wheels is now given by 

P=^-> (145«0 Art. 1787. 

and the pitch of wheel vanes by 

P 1 = ^p. (146*0 Art. 1787. 

The pitch at the outflow ends of guide vanes for radial 
flow turbines is 

^=^P- (145*0 Art. 1787. 

For the inflow ends of the wheel vanes the pitch is 

P x =."-£±. (146*0 Art. 1787. 

Width of Vanes. — Width of outflow end of guide 
vanes, 

(147.) Art. 1795. 



Zx-ZJ 

The width e 1 of the inflow end of the wheel vanes is made 
a little greater than e, usually 

* 1 = * + ±inch to <? + £. (148.) Art. 1796. 

Width of outflow end of wheel vanes, 
Q 



z , x , v , 



(149.) Art. 1796. 



132 TABLES AND FORMULAS. 

FORMULAS USED I1V HYDRAULIC MACHINERY. 



SIZE OF AIR AND VACUUM CHAMBERS. 

Let V = volume of piston displacement ; 
V t = volume of air chamber; 
V n — volume of vacuum chamber. 
For ordinary double-acting pumps working under moder 
ate pressures at ordinary speeds, 

V 1 = 3V. Art. 1885. 

For pressures of 100 pounds per square inch and upwards, 
or for high piston speeds, 

V 1 = 6V. Art. 1885. 

For ordinary cases, make 

K = iV t - Art. 1889. 



CALCULATIONS RELATING TO PUMPS. 



Displacement. 

Let D = displacement in cubic feet per minute; 

d = diameter of piston or plunger in inches; 

L = length of stroke of piston or plunger in inches; 

N= number of discharge strokes made by piston or 
plunger in 1 minute. 
Then, D = . 000455 d 2 L N. Art. 1 905. 

Slip. 

Let s = slip; 

D = displacement ; 
C — actual discharge. 

Then, S= D B ^ ' • Art. 1909. 

Head and Pressure. 

Let H — head in feet; 

P = pressure in pounds per square inch. 

P=,ASkH. Art. 1914. 

#=2.304 P. Art. 1915. 



TABLES AND FORMULAS. 133 

Size of Piston or Plunger. 

Let G = number of gallons discharged per minute; 

5 = speed in feet per minute of piston or plunger; 

d = diameter of piston or plunger in inches; 

F = number of cubic feet discharged per minute. 
Then, the theoretical diameter of piston or plunger is 

^/= 4.95 V £ = 13.54 y~. Art. 1916. 

If we add 25 per cent, to the required discharge to allow 
for slip, the diameter of the piston or plunger will be 

d=5.535-v£- = 15.ldSy^r. (152.) Art. 1916. 

Discharge. 

The theoretical discharge in cubic feet equals the displace- 
ment. 

The theoretical discharge in gallons per minute is 

G = . 0408 d*S. Art. 1917. 

If we make the same allowance for slip as was made in 
formula 152, the discharge in gallons per minute is 
G= .03264^ 2 5; (153.) Art. 1917. 

and in cubic feet per minute, 

F= .00±36d 2 S. Art. 1917. 

Power. 

Let H. P. = horsepower; 

H = vertical height in feet from the surface of the 

water in the well or sump to the center of the 

outlet end of discharge pipe ; 
G = discharge in gallons per minute; 
F = discharge in cubic feet per minute* 

The theoretical power is 

H. P. = .000254 GH= .0019 F H. Art. 1918. 

If, for ordinary cases, the factional resistances are 



134 TABLES AND FORMULAS. 

assumed to be 50 per cent, of the power developed by the 
engine, the power required is 

H. P. = .00038 G H. (154.) Art. 1918. 

To find the height to which a given power will raise a 
given quantity of water, making the same allowance for 
friction as in the last formula, 

Size of Steam Cylinder. 

Let J> = steam piston speed; 

d = diameter of steam cylinder in inches ; 

r = ratio between length of stroke and diameter of 

cylinder; 
/ = length of stroke in feet; 
N ■= number of strokes per minute; 
H. P. = horsepower; 

P — steam pressure in pounds per square inch. 

Then, for simple direct-acting steam pumps, 

= V — (156.) Art. 1920. 



'504,201.6 X H. P. 


rPN 


'42,01.6.8 X H. P. 



or, d = \ — i — ^C — . (157.) Art. 1920. 

Having obtained the diameter of the steam piston by 
either of the above formulas, the stroke can be found by 
multiplying the diameter by the value of the ratio r. When 
formula 157 is used, the number of strokes can be found 
by dividing the piston speed by the length of the stroke 
in feet. 

Sizes of Suction and Delivery Pipes. 

For a velocity of 200 feet per minute in the suction pipe 
and 400 feet per minute in the delivery pipe, 

Let d z — diameter of suction pipe ; 
d 2 = diameter of delivery pipe ; 
G =k discharge in gallons per minute. 



TABLES AND FORMULAS. 135 



4 = 4.95V^r, ov_d x = .U^G. (158.) Art. 1921. 

< = L95j/^-, or d t = .^6)/G. '(159.) Art. 1921. 

The pipes may be made larger than the values calculated 
by the above formulas, particularly the suction pipe, but it 
is not good practice to make them any smaller. 



DUTY OF A PUMP. 

Old Standard. — According to the old standard, the duty 
of a pumping engine is the number of pounds of water raised 
one foot high for each 100 pounds of coal burned in the boiler. 

Let G = number of gallons discharged in a given period; 

h = total vertical distance in feet from the surface 

of the water in the well, or other source of 

supply, to the point of discharge; 

W= the number of pounds of coal burned in the 

given period ; 
D = the duty in foot-pounds. 
Then, 

I)= S35 f v G/ \ (160.) Art. 1922. 

Standard Recommended by Committee of Ameri- 
can Society of Mechanical Engineers. 

The duty of a pumping engine is equal to the total 
number of foot-pounds of work actually done by the pump, 
divided by the total number .of heat units in the steam used 
by the pump, including the steam used by the condensers 
(if any) and boiler feed, and this quotient multiplied by 
1,000,000. 

The number of foot-pounds of work done by the pump is 
to be found as follows: A pressure gauge is attached to the 
discharge pipe and a vacuum gauge to the suction pipe, 
both as near the pump as convenient; then the pressure 
against which the pump plunger works is equal to the 
difference in the pressures shown by these two gauges plus 

G. G. IV.— SI 



136 TABLES AND FORMULAS. 

the head due to the difference in level of the points in the 
pipes to which they are attached; and the number of foot- 
pounds is equal to the continued product of the net area of 
the plunger (making allowance for piston rods), the length 
of the plunger stroke in feet, the number of plunger strokes 
made during the trial, and the pressure against which the 
pump plungers work, as shown by the gauges. 

The number of heat units furnished to the pump is the 
number of British thermal units (B. T. U.) in the steam 
from the boilers, and is to be determined by an evaporation 
test of the boilers. If we let 

A ="the net area of the plunger in square inches; 

P = the pressure in pounds per square inch indicated by 

the gauge on the discharge pipe; 
p = the pressure in pounds per square inch corresponding 
to the vacuum indicated by the gauge on the suction 
pipe; 
vS = the pressure in pounds per square inch corresponding 

to the difference in level between the two gauges; 
L = the average length of stroke of pump plunger in feet ; 
N = the total number of single strokes of plunger made 

during the trial; 
i/=the total number of heat units consumed by the 

engine during the trial; 
W= the total number of foot-pounds of work done by the 

pump during the trial; and 
D — the duty. 
Then, W=A{P±p + S)LA\ (161.) Art. 1924. 

W 

and D = -rr X 1,000,000 = 

Ii 

A ( P ±P + S ) LN x 1,000,000. (16.2.) Art. 1924. 

CALCULATIONS RELATING TO HYDRAULIC MACHINERY. 



Relations Between Pressure and Size of Ram. 

Let D = the diameter of a hydraulic piston or ram ; 

IV = the weight of the ram and attachments that 
must be lifted by the water; 



TABLES AND FORMULAS. 137 

p = the pressure of the water in pounds per square 

inch ; 
F= the percentage of friction; 
P = the net pressure exerted by the ram. 
To find the net pressure exerted by a ram or plunger of a 
hydraulic press, 

P=.7854X^ 2 X/X (l-T^)- W. (163.) 

V iUU/ Art. 1969. 

To find the pressure per square inch required to exert a 
given net pressure when the diameter and weight of the 
ram and the percentage of friction are given, use the for- 
mula 

P-\- W 
p= — ^ -T-. (164.) Art. 1970. 



7854 D 2 x 



(>-i£) 



To find the diameter of piston or ram required to exert 
a given net pressure, use the formula . 



D= /— i-±— _. (165.) Art. 1971. 

y.7854x/x(l--^ 

'Weight and Volume of Accumulators. 

Let W x = weight of accumulator ram ; 
W 2 = load on accumulator ram ; 
D i = diameter of accumulator ram; 
p\ = maximum pressure per square inch in the accu- 
mulator cylinder; 
p 2 = minimum pressure per square inch in the accu- 
mulator cylinder; 
p = mean pressure per square inch in the accumu- 
lator cylinder; 
.S = stroke of accumulator ram ; 
V = total volume of water displaced by accumulator 

ram during the stroke S; 
F — the percentage of friction. 
To find the mean pressure p corresponding to a given 
case, use the formula 



138 TABLES AND FORMULAS. 

W 4- W 

f = .nuD* ' (166 - } Art - 1973 - 

The maximum pressure is found by the formula 

A = — 1Vl + \ V ' 2 zTx, (167.) Art. 1973. 



7854 D" x 



\ 100/ 



and the minimum pressure by 

W J- IF 

-p-. (168.) Art. 1973. 



7854 D; x 



>+« 



The weight required to produce a given mean pressure 
when diameter and weight of the ram are known may be 
found from the following formula: 

W 9 = .7854 x^'X/- ^v (169.) Art. 1974. 

The relations between the stroke, diameter, and volume 
of an accumulator are given by the following formulas: 

F =.7854 -A 2 5. (170.) Art. 1977. 
S-^UJJ. (171.) Art. 1977. 

^ = 1.128'/-^. (172.) Art. 1977. 

In the above formulas, if D x and S are in inches, the 
volume will be given in cubic inches ; and if D 1 and vS are in 
feet, V will be given in cubic feet. 



FORMULAS USED IjV WATER SUPPLY AIVD 
DISTRIBUTION. 



DIMENSIONS OF SPILLWAY OR OVERFLOW. 

Let L = length of lip of spillway in feet; 

A = area of watershed above dam in square miles ; 
D — depth of notch of spillway in feet ; 
Q = cubic feet of water per second per square mile; 
C = a constant depending on the character of the 
dam and its surroundings and the area of the 
watershed. 



TABLES AND FORMULAS. 130 

Then, L = 20 \/A. (173.) Art. 2048. 

D = ^~~~ +C. ( 1 74.) Art. 2048. 

If we assume Q = 64, which corresponds to a little over 
41 million gallons per 24 hours, per square mile, and repre- 
sents a very powerful freshet flow, although, perhaps, not 
the maximum, formula 174 reduces to 

D = VA + C. (175.) Art. 2049. 



MASONRY DAMS. 

Let A = thickness of top of a trapezoidal dam in feet; 
B = thickness of base of dam in feet ; 
C — a factor of safety against either sliding or 

overturning; 
D = density (weight per cubic foot) of material of 

which dam is built; 
H = head of water pressing against the dam in feet ; 
R — resistance of wall to sliding; 
T — horizontal thrust in pounds on the dam, due to 
the head H\ 
MR = moment of resistance of dam against over- 
turning by rotating about its outer toe; 
MT= moment of thrust about the outer toe of the 
dam. 
The thrust is 

T= 31.25i/ a , (176.) Art. 2063. 

and the moment of thrust 

MT=10A2H\ (177.) Art. 2063. 

The resistance of the wall to sliding is 

R = 0.75.4 D. (178.) Art. 2065. 

The moment of resistance to overturning for a wall with 
vertical sides is 

MR = -4rA (1 79.) Art. 2066. 



140 TABLES AND FORMULAS. 

and for a trapezoidal wall 

MR = ^(aB-^-+b\ (180.) Art. 2066. 

The relation between A, B, D, and H for a factor of 
safety C against sliding is given by the formula 

B = S333 I) CI1 -A. (181.) Art. 2067. 

For a factor of safety C against overturning, the breadth 
of the base is given by the formula 



B = i y / ^oC_H^ + 3A ,_A (182>) Art 2068 

Average Dimensions. — For practical values of A and 
D, a satisfactory value of B is 

B = iH to \H. Art. 2070. 



HIGH MASONRY DAMS. 



Maximum Unit Stress on Base of Dam for 
Unequally Distributed Load. 

Let L = length of base of a section through the dam; 
d = length of the shorter segment of this base; 
L — d— length of the remaining segment; 
W — the resultant of the weight of the section, or 

the vertical component of this resultant; 
P = maximum unit stress. 

There are three empirical formulas for the value of P, 
which experience shows give satisfactory results; viz. : 

P=^ (L-1.5d), (183.) Art. 2072. 

2 W 
P=^-r, (184.) Art. 2072. 

o cl 

and P= n/( ^7' - (185.) Art. 2072. 

Of these formulas, the last is probably the most satis 
factory. 



TABLES AND FORMULAS. 141 

DARCY'S FORMULAS. 

Let D = diameter of pipe in feet; 
i/= total head in feet; 
L = total length in feet; 
V •=■ velocity of efflux in feet per second; 
C = an experimental coefficient (see table of Coeffi 

cients for Darcy's Formula) ; 
Q — quantity discharged in cubic feet per second ; 
A = area of pipe in square feet; 

Ti- 
ll — head per 1,000 feet of length = -^-. 

Fundamental Formulas for Long Pipes. 
1. (186.) Art. 2092. 



CL V" 

C 



V^V^Jf. (187.) Art. 2092. 



Q = A\/^Jf. (188.) Art. 2092. 



C L 



Q = 0.7854: D*y^^. (189.) Art. 2092. 

Q = l^ ' 61 ^ — • (190.) Art 2092. 

Approximate Formulas for Rough Pipes. 

For pipes from 8 inches to 48 inches in diameter, 



Q 


-A 


000 D" H 


L 




Q = 


\/Wli. 




Q 2 

h IT 


= l. 




Q = n 


12 \/J)li. 




Q 


- l. 



D'\/Dk 



(191.) Art 2094. 
(192.) Art 2094. 
(193.) Art 2094. 

(194.) Art. 2094. 
(195.) Art. 2094. 



14-2 TABLES AND FORMULAS. 

For pipes from 3 inches to 6 inches in diameter, 
j2jf= 0.785. (196.) Art. 2094. 

Q = 0.S9 i f^h, (197.) Art. 2094. 

Formulas for Smooth Pipes. 



Q=\/ZD*h. (198.) Art. 2095. 

j^=%- (199.) Art. 2095. 



Q = 1A0 iWUi. (200.) Art. 2095. 

General Relation Between Smooth and Rough 

Pipes. 

In general, the K discharge through a smooth pipe is l.Jfi 
times that t /trough a rough pipe of the same diameter; and, 
reciprocally^ the discharge through a rough pipe is <>. 70 times 
that through a smooth one of the same diameter. These 
factors represent the practical limits between which the 
extremes of smoothness and roughness can affect the flow. 
Art. 2095. 

Formulas for Velocity. 

For rough pipes of from 8 inches to 48 inches in diameter, 

V=l.%1i/j5J. (201.) Art. 2096. 

For rough pipes of smaller diameter, 

V= 1.13 j/SJ. (202.) Art 2096. 

For smooth pipes of large diameter, 

P=1.78y7?A. (203.) Art. 2096. 

For smooth pipes of small diameter, 

V = l.m\/l)Ji. (204 ) Art. 2096. 



TABLES AND FORMULAS. 143 

General Relation Between the Elements of Two 

Pipes. 

Let D } Q, Z, //, and C be the respective elements of one 
pipe and D' , Q', L\ H\ and C the similar elements of 
another; then, 

DHC'L'V* _ 

D'H'CL F 2 ~ 

If, as can usually be done, we make C= C, we have 
DHL'V" 



D'H'L V 



i = l. (205.) Art. 2097. 



Also, Q q^ d ^ h = 1- (206.) Art. 2097. 

If L and H equal, respectively, L and U', 

^y = K^t- (207.) Art. 2098. 

To find the number x of small pipes with the diameter D' 
to replace a pipe whose diameter is D, 



— r n'5 • 



-^. (208.) Art. 2098. 



COMPOUND PIPES. 

To find the diameter of a simple pipe that will give the 
same delivery as a given compound system : 

Let D = diameter of the simple pipe ; 
L = length of the simple pipe ; 
d, d', d", etc. — diameters of the respective sections 

of the compound pipe; 
/, /', /", etc. = lengths of the respective sections of 

the compound pipe. 

Then, 
l p , = y + jn + j„ B + etc. (209.) Art. 2111, 



U± TABLES AND FORMULAS. 

PUMPING INTO MAINS. 

Theoretical horsepower required to force a given quantity 
of water into a main against a given pressure head : 

Let H. P. = theoretical horsepower; 
H — pressure head in feet ; 
Q = quantity of water in cubic feet per second. 

H. P. =%^. (210.) Art. 2117. 

o. o 



WEIGHTS AND THICKNESS OF CAST-IRON PIPES. 

Let W = weight in pounds; 
D — diameter in inches; 
T — thickness in inches; 
L = length in inches; 

P — weight in long tons (2,240 pounds) ; 
M = length in miles; 

W = approximate weight per foot in pounds; 
H = total head in feet. 

IV=0.SZ(D+ T)TxL. (211.) Art. 2125. 

W = 10(D+ T) T. (212.) Art. 2126. 

P=Z5M(D+T)T. (213.) Art. 2127. 

r= 0.00006 HD + 0.01S3D + 0.296. (214.) Art. 2128. 



DARCY'S FORMULAS FOR FLOW IN OPEN CHANNELS. 

Let U = mean velocity of now in feet per second; 
5 = water section in square feet ; 
WP= wet perimeter in feet; 

R = mean hydraulic radius = • 

I V A 

f= slope of free water surface per foot of length = 

total fall of surface divided by total length; 
D = interior diameter of a circular conduit in feet. 



TABLES AND FORMULAS. 145 

For an ordinary tunnel or channel lined with well-laid 
brick. 



For a circular brick-lined conduit running full, 



"-&£$&* «■«•>*«•*«* 



FORMULAS USED IN IRRIGATION. 



APPROXIMATE DISCHARGE OF WEIRS. 

Let / = length of notch in feet; 

H = measured head on crest in feet; 
Q = discharge in cubic feet per second; 

then, Q=3ilH\ (217.) Art. 2163. 



FLOW OF WATER THROUGH CONDUITS. 

Let /* = difference in level between the ends of the canal, 
or any two cross-sections of the canal; 
/ = horizontal length of that portion of the canal 
included between the sections whose difference of 
level is k; 

. // 
s = slope = the ratio y; 

a = area of the water cross-section; 
p = wetted perimeter; 

r — hydraulic radius = the ratio — ; 

P 
c' = sl coefficient depending on the nature of the sur- 
face of the conduit; 
and v = mean velocity of flow. 

The laws for the resistance to flow may be expressed by 
the relation h a — c' I p v\ from which we have the general 
formula 



v = \ 4 X t = Vn x s x r - (218.) Art. 21 73 

c L p c 



14fi TABLES AND FORMULAS. 

By replacing \ — by the equivalent factor c, we have 
v — c \/r s^ the same as formula 50, Art. 1033. 
Formulas for Flow in Canals. 

Canals with earthen banks, 



,7100,000 r 2 .y ln ^^s a ^.~~ 

v = y ' (219.) Art. 2177. 

9 r -t- do 



Canals lined with dry stone, 

(220.) Art. 2183. 



.7100,000 r*s 



8r+ 15 
Canals lined with rubble masonry 



-/» (221.) Art. 2184. 

Wooden flumes, 



sm^- «»•> *•• *>»<>■ 



TIMBER FOR FLUMES. 

Let JV = total load in pounds carried by any beam; 
/ = length of beam in inches; 
I? — breadth of beam in inches; 
d = depth of beam in inches ; 

S = maximum unit fiber stress in pounds per square 
inch. 

For a simple beam with a uniformly distributed load, 
W=i b -^-S. (222.) Art. 2189. 

For a simple beam with a concentrated load at the middle, 
JV=i^-S. (223.) Art. 21 90. 



TABLES AND FORMULAS. 



147 



For a beam with a concentrated load at a distance l x from 
one support and / 2 from the other, where l x -\- 1 2 = /, 

bdHS 



W=l 



T /./„ 



(224.) Art. 2191. 



For a beam on which the load at one end is zero, with a 
uniform increase in the load to the other end, 

lV=1.3 l ^p-S. (225.) Art. 2192. 



SAFE WORKING STRESS S. 

For good sound timber. 



Kind 


of Timber. 




Safe Working Stress. 


Steady Load. 


Variable Load. 


Yellow Pine 


i,8oo 
h 2 5° 

1,200 
I, IOO 


I, 200 


White Oak 


1,000 


Spruce 


9OO 
850 
SOO 


Hemlock 


White Pine 










TRUSSES. 





Trussed Stringers. 

Let L = span in inches; 

H = depth of truss in inches; 
/; = breadth of stringer in inches; 
d = depth of stringer in inches ; 
W t = total uniformly distributed load in pounds, 
5 = allowable unit fiber stress in stringer or strut; 
S 2 = total stress in tie-rods; 
h = width of strut in inches; 
t — thickness of strut in inches. 



148 TABLES AND FORMULAS. 

Relation between W t and dimensions of stringer, 

Stress in tie-rods. 



^ = fV^/(y)+^ 2 . (228.) Art. 2198. 

Stress in strut, 

W t = \htS. (229.) Art. 2198. 

The King-Rod Truss. 

Let L = length of span in inches; 

W t — total uniformly distributed load in pounds; 
W = total stress in each strut in pounds; 
S a = safe unit stress in king-rods ; 
A = net sectional area of king-rods; 
and H — depth of truss in inches; 



then, IV=^^Y^\'+H\- (230.) Art 2199. 

and A=f-^-. (231.) Art. 2199. 

The Queen-Rod Truss. 

Let S t = maximum unit stress in tie-beam; 

JV = total uniformly distributed load in pounds; 
L = length of span in inches; 
H = depth of span in inches ; 

b = breadth of tie-beam in inches; 

d = depth of tie-beam in inches; 

S c = total stress in upper chord member in pounds; 

.S s = total stress in struts in pounds ; 

S n — allowable unit stress in queen-rods; 

A = sectional area of queen-rods. 

The maximum unit stress in tie-beam is 
S '=- ] -J7(ui+^)- < 232 -> Art.2300. 



TABLES AND FORMULAS. 149 



The total stress in the upper chord member is 
S c = i ^. (233.) Art. 2200. 

The total stress in each strut is 



S.=±WVl + itjp- (234.) Art. 2200, 

The net area of each queen-rod is 

W 
A = i -£-. (235.) Art. 2200. 

The Howe Truss. 

Let N t = the number of a tie, counting from the center; 
S t = total stress in a tie ; 
L t = length of a tie ; 
L s = length of a strut; 
S a = total stress in a strut; 
N = number of panels in truss from center to either 

abutment; 
n = number of panels from a given panel to the 

nearer abutment ; 
P = panel load in pounds ; 
L p = length of a panel ; 

S tc = total stress in top chord of a given panel ; 
S bc = total stress in bottom chord of a given paneL 

The stress in any tie whose number is N t is 

St = W + i) P- (236.) Art. 2202. 

The stress in a strut is 

S s = ^S t . (237.) Art. 2202. 

The top chord stress is 

S t0 = nP{N-in)^. (238.) Art. 2202. 



150 TABLES AND FORMULAS. 

The bottom chord stress is 
S bc =P(Nn+N-n-iri 2 -i)^-. (239.) Art. 2202. 



REFUSAL OF PILES. 

Let 5 — weight a pile will bear with safety; 

IV = weight of hammer, in the same unit as S 
H — height of fall of hammer in feet; 

then, 5= W H. (240.) Art. 2205o 



INDEX. 



TABLES. page 

Common Logarithms . . i-iq 

Natural Sines, Cosines, Tangents, 

and Cotangents . . . 21-40 

Traverse Tables . „ . . 41-48 

Resisting Values of Materials . 49, 50 
Horizontal Distances and Differ- 
ences of Elevation for Stadia 
Measurements 
Radii and Chord and Tangent De 

flections 

Moments of Inertia . 

Bending Moments and Deflections 

Specific Gravities and Weights per 

Cubic Foot .... 
Discharge of Standard Orifices 

" " Weirs .... 

Coefficients of Friction for Smooth 
Cast or Wrought Iron 
Pipes . 
" for Angular Bends 

" " Circular Bends 

" " Darcy's Formula 

Properties of Saturated Steam 
Standard Dimensions of Wrought 
Iron Steam, Gas, and 
Water Pipes . 
" Pipe Flanges . 
Specific Heat of Substances . 
Constants for Apparent Cut-Offs 

Used in Determining M. E. P 
Riveted Joints of Boilers . 
Positions of Eccentric Relative to 

Crank .... 
Diameters of Steam and Exhaust 

Pipes 

Piston Speeds of Steam Engines 
Ratio of Grate Area of Boiler to 
Horsepower . 
44 " Heating Surface to Grate 

Area ... 
" " Heating Surface to Horse 
power ... 
Factors of Evaporation . 
Size of Chimneys and Horsepower 
of Boilers 

G. G. IV.— 32 



So 



RULES AND FORMULAS, page 
Formulas Used in Algebra . . 83 
Trigonometric Functions . . 83 

Rules for Using Tables of Loga- 
rithms of Num- 
bers . . 84-86 
" " " Trigonometric 

Tables . . 86 



51-00 


Rules for Mensuration. 




61-63 

64 

5 65 


The Triangle 

" Quadrilateral .... 


87 
87 


44 Circle 


87 


" Ellipse 


88 


66 68 


" Prism and Cylinder 


88 


68-69 
70 


" Pyramid and Cone 

" Frustum of a Pyramid or 


89 


Cone . 


8q 


1 


" Sphere 


89 


71 


Formulas Used in Elementary 




72 


Mechanics. 




72 


Uniform Motion 


80 


72 


Mass, Weight, and Gravity . 


90 


73-75 


Formulas for Gravity Problems 


90 




Falling Bodies .... 


90 




Centrifugal Force .... 


qi 


76 


Center of Gravity of Two Bodies 


92 


77 


The Efficiency of a Machine 


02 


78 


Work 


92 




Power 


Q2 


78 


Kinetic Energy .... 


92 


78 


Density 


93 



Rules and Formulas Used in 
Hydraulics. 

Pascal's Law 93 

General Law for the Downward 
Pressure Upon the Bottom of 

Any Vessel 93 

General Law for Upward Pres- 
sure 93 

General Law for Lateral Pressure 04 
44 4l ' 4 Pressure . . 94 

Specific Gravity . . . .94 
Mean Velocity .... 95 

Velocity of Efflux from an Orifice 95 



VI 



INDEX. 



Rules and Formulas Used in 

Hydraulics— Continued. page 

Theoretical Range of a Jet . . 96 

Velocity of a Jet .... 96 

Discharge of an Orifice ... 97 

" " Standard Orifices . 97 

" " a Submerged Rect 

angular Orifice 
" " Weirs 

Flow of Water Through Pipes 99-102 
" " Water Through Conduits 

and Channels . 102-104 
Values of the Coefficient of Rough- 
ness for Use in Kutter's For- 
mula 103 

Formulas Used in Pneumatics. 
Pressure, Volume, Density, and 
Weight of Air When the Tem- 
perature Is Constant . . .104 
Mariotte's Law . . . .104 
Pressure and Volume of a Gas 

with Variable Temperature . 105 
Gay-Lussac's Law . . . .105 
Mixture of Two Gases Having 

Unequal Volumes and Pressures 106 
Mixture of Two Volumes of Air 
Having Unequal Pressures, Vol- 
umes, and Temperatures . . 106 

Formulas Used in Strength of 
Materials. 
Unit Stress, Unit Strain, and Co- 
efficient of Elasticity . . .106 
Strength of Pipes and Cylinders 107 
Moment of Inertia, Resisting Mo- 
ment, and Bending Moment of 

Beams 107 

Deflection of a Beam . . .108 
Strength of Columns . . .108 
'* " Shafts .... 109 

Constants for Shafting . . .no 
Strength of Ropes and Chains . no 

Formulas Used in Surveying. 

Radius of a Curve . . . .in 

Length of Subchords . . . m 

Length of a Tangent of a Curve . m 

Chord Deflection . . . .112 

Tangent Deflection . . .112 

Stadia Measurements . . .112 

Barometrical Leveling . . .113 

Rules and Formulas Used in 
Surveying and Mapping. 

Rule for Balancing a Survey . 113 
" " Double Longitudes . 113 

Application of Double Longi- 
tudes to Finding Areas . .114 



Rules and Formulas Used in 
Surveying and Mapping— 

Co n tin ued. page 

Trapezoidal Rule . . . .114 

Simpson's Rule . . . .114 

Volumes of Irregular Solids . 115 

The Prismoidal Formula . . 115 

Latitudes and Departures . . 115 

Formulas Used in Steam and 
Steam Engines. 

Specific Heat 115 

Temperature of Mixtures . .116 
Mixture of Steam and Water . 116 
Work Done by Piston . . .116 
Real and Apparent Cut-Off . . 116 

Horsepower 117 

Mean Effective Pressure . .117 

Piston Speed n 7 

Mechanical Efficiency of Engine 117 
Steam Consumption . . .117 
Thermal Efficiency of Engine . 118 
Water Required by Condenser . 118 
Ratio of Expansion . . .118 

Formulas Used in Steam Boil- 
ers. 
Air Required for Combustion 

and Heat of Combustion . . 119 
Strength of Boiler Shells . . 119 
Horsepower of Boilers . . .119 

Safety Valve 120 

Draft Pressure of Chimney . . 120 
Quality of Steam . . . .121 

Formulas Used in Water-Wheels. 
Theoretical Energy of a Given 

Head and Weight of Water . 121 
Theoretical Power . . . 121 

Energy of a Jet . . . .122 
Pressure Due to Impact and Re 

action of a Jet 
Efficiency .... 
Overshot Water- Wheels 
Breast Wheels 
Undershot Wheels . 
Poncelet's Wheel . 
Turbines .... 



IN 



122 
124 
1^4 
125 
125 
125 
126-131 

Hydraulic 



Formulas Used 

Machinery. 
Size of Air and Vacuiim Cham 

bers 

Displacement of Pumps 

Slip 

Head and Pressure 

Size of Pump Piston or Plunger 

Discharge of Pumps 

Power of Pumps . 



132 
132 
132 
132 
133 
133 
133 



INDEX. 



Vll 



Formulas Used in Hydraulic 
Machinery— Continued. page 
Size of Steam Cylinder for Pumps 134 
Sizes of Suction and Delivery- 
Pipes 134 

Duty of a Pump . . . .135 
Relations Between Pressure and 

Size of a Ram .... 136 
Weight and Volume of Accumu- 
lators 137 

Formulas Used in Water Sup- 
ply and Distribution. 
Dimensions of Spillway or Over- 
flow 138 

Masonry Dams . . . .139 

High Masonry Dams . . .140 

Darcy's Formulas for Long Pipes 141 

" " " Rough Pipes 141 

" " " Smooth Pipes 142 

General Relations Between 

Smooth and Rough Pipes . 142 



Formulas Used in Water Sup- 
ply and Distribution— Cont. page 
Darcy's Formulas for Velocity . 142 
General Relation Between Ele- 
ments of Two Pipes . . .143 
Compound Pipes . . . .143 
Pumping Tnto Mains . . .144 
Weights and Thickness of Cast- 
iron Pipes . . . . .144 
Darcy's Formulas for Flow in 
Open Channels . . . .144 

Formulas Used in Irrigation. 
Approximate Discharge of Weirs 145 
Flow of Water Through Conduits 145 
" in Canals . . . .146 
Timber for Flumes . . .146 
Trussed Stringers . . . .147 
King-Rod Truss . . . .148 
Queen-Rod Truss . . . .148 
Howe Truss . 149 

Refusal of Piles . . . .150 



ONE COPY F 

JUL 15 1904 



13 1904 



H<6l 22 VW 



